The Trump administration is requiring states to change the way they conduct elections or risk losing tens of millions of federal terrorism-prevention funds, in its latest move that would make voting harder and undermine trust in results that don’t go President Trump’s way.

The effort would force states to transition to paper ballots, verify citizenship of voters and make other changes to election procedures, according to Federal Emergency Management Agency documents.

FEMA, part of the Homeland Security Department, has told states that it would withhold 20 percent of some terrorism-preparedness grants unless they provide “proof of compliance” with the election security measures, the documents show. The grants, which total $1 billion each year, help pay for physical barriers and other security measures as well as planning and drill exercises and cybersecurity protections.

Courts have already blocked similar efforts by the administration to require changes to voting, saying the Constitution grants power over elections to states and not the executive branch.

Mr. Trump and his allies have made false claims of voter fraud for years, which have been investigated and debunked. But since he returned to power last year, he has tried to use the levers of the federal government, along with his influence over state and local lawmakers, to reshape American elections, though with little success. Mr. Trump has pushed new policies through cabinet agencies, including the Justice Department and the Homeland Security Department, and drafted two executive orders seeking to make extensive changes to how elections are run. Both of those executive orders have largely been blocked by courts.

Homeland Security officials said in an unsigned statement that election security and integrity were top administration priorities and that grant recipients should face accountability.

The policies align closely with some of the goals pushed by right-wing activists that are rooted in false conspiracy theories about election machines and voting by noncitizens during the 2020 election.

David Becker, executive director of the Center for Election Innovation and Research in Washington, said he expected the rules to fall if they are challenged in court, adding that they would not increase election security.

“They will actually harm election security,” Mr. Becker said. “These are radical changes to election processes.”

FEMA announced the annual antiterrorism grants on June 24, saying projects to be funded should align with priorities including “protecting the integrity of American elections” and “supporting border security efforts.” A notice warned recipients that 20 percent of any award could be withheld until the Homeland Security Department verifies compliance with the new rules.

The rules do not apply to federal disaster aid, which FEMA also manages.

CNN previously reported on the administration’s plans to tie election security rules to the homeland security grants.

The grants go largely to populous states with large urban areas, including New York, California and Texas. New York is set to receive about $204 million in fiscal year 2026, according to the office of Gov. Kathy Hochul.

“After denying disaster relief funding and stripping millions from counterterrorism programs, the Trump administration is once again putting New Yorkers’ lives at risk to forward their political agenda,” Ms. Hochul, a Democrat, said in a statement about the new requirements. “Unlike the President, my number one priority is New Yorkers’ safety, and I will fight to ensure our state has every resource available to keep us secure.”

The changes could face numerous legal roadblocks, partly because the executive branch does not have constitutional authority to regulate elections, which states administer and Congress broadly oversees.

Last month, a judge barred an effort by the Trump administration to force states to use a centralized national database of citizens built for checking immigration status, known as the SAVE system, to verify voters on their rolls, saying the move violated at least three laws. Two provisions in the FEMA grants would require the same process, according to internal FEMA documents obtained by The New York Times.

The internal documents also show that states would be required to submit plans and timelines to transition from systems that use bar codes or QR codes to rely solely on hand-marked paper ballots. Another provision calls for grant recipients to provide proof that they are conducting manual post-election audits of 5 percent of ballots to ensure electronic systems accurately count votes.

Switching to hand-marked paper ballots would very likely cost states hundreds of millions of dollars, as they would be required to buy new tabulation machines and other election equipment. Hand-marked paper ballots would still require the use of machines to tabulate votes. Nearly every state already has some form of paper ballot backup for audits and tabulation.

Manual audits of 5 percent of ballots would also quite likely to cause significant delays in counting, cost millions of dollars and, in some cases, fall far short of what would be considered an adequate audit for races with narrow margins, as is often the case in some congressional districts and local elections, Mr. Becker said. Currently, every state has a post-election audit process to check for accuracy, according to the National Conference of State Legislators.

The number of students admitted to Ph.D. programs this fall dropped 15 percent from the previous year, according to data from over 50 top research universities, raising fears that the nation’s capacity to produce new science could be diminished.

The decline is driven, in part, by a chaotic and unpredictable federal funding environment under the Trump administration, as federal cuts are promised and then reversed, and budgets remain unclear.

A reduction in doctoral students could mean fewer scholars at universities to teach and mentor undergraduates. Higher education leaders also worry that, if the declines continue, there will be fewer researchers to power a rapidly evolving scientific work force.

The data showing the decrease comes from 55 universities, all of them members of the Association of American Universities, an invitation-only organization that includes 69 of the most prestigious research institutions in the United States. The data collection was conducted by a another group, the Association of American Universities Data Exchange.

Schools in A.A.U. confer half of the nation’s research doctorates, according to the association.

“We are at risk of losing a whole generation of new talent because of the reduction in the capacity to support those students,” said Toby Smith, vice president for policy of the A.A.U.

University leaders and research advocates cite many reasons for the declines in new doctoral students. Key federal agencies, such as the National Institutes of Health and the National Science Foundation, have been funding fewer research grants. The wealthiest institutions also face a new federal tax on their endowments.

But the most cited reason in interviews was the unreliable nature of federal funding under the Trump administration. The administration proposed major cuts to federal research agencies last year, but Congress restored the funding. It is again proposing big cuts. While Congress may again reverse the administration’s proposed reductions, the uncertainty makes it hard for schools to make multiyear commitments to doctoral students.

The administration also abruptly ended thousands of research grants last year, arguing that they did not align with the government’s priorities. The administration restored many of the grants after judges deemed the eliminations illegal and arbitrary, but research advocates say the whiplash was damaging.

Hiring for research projects is not like turning a light switch on and off, they said, and disruptions can have a long tail.

Mr. Smith said declines in funding at agencies like the National Science Foundation, where new awards have “dwindled to a trickle,” had disproportionately hurt new scholars hoping to start their scientific careers.

Travis York, a director at the American Association for the Advancement of Science, said developing research talent “is more like planting an orchard than filling a warehouse.”

Dr. York, who has a doctorate in higher education administration, said the consequences of a decline in new Ph.D. students because of funding uncertainty “will not simply be felt on campuses this fall and next year; it really will ripple across the U.S. innovation ecosystem for years to come.”

In a statement, Liz Huston, a White House spokeswoman, said President Trump was “committed to keeping America the world leader in scientific innovation.”

“That means funding real science, not left-wing ideology,” she said. “For years, federal grant dollars were funneled into politicized research with little accountability to taxpayers or results to show for it. This administration is ending that waste and restoring a simple standard: Grants should go to the best science, not politics. That’s how we maintain American global dominance.”

The data provided to The New York Times show only overall reductions and do not show breakdowns by discipline. But anecdotal data and announcements from several universities suggest they are hitting science and humanities programs across the board.

Declines this fall also appear to be stacking on top of losses in Ph.D. student capacity from last year. The A.A.U.D.E. collected data for fall 2025 and found that, for the 42 schools that responded, new enrollments dropped by 11 percent from the previous year.

Enrollments and admissions are not the same, but tend to be good proxies for each other in doctoral education. The central takeaway, A.A.U. officials said, is that the data shows two years of a “substantial reduction in the number of Ph.D. students being admitted and ultimately enrolled at major research universities.”

Sally Kornbluth, the president of the Massachusetts Institute of Technology, said in a message to the campus in May that, while Congress restored some agency cuts, the money was “not actually flowing to M.I.T. the way it typically has.” She said new federal research awards were down 20 percent and also cited the effects of a new tax on the school’s endowment.

The school is expecting nearly 20 percent fewer new graduate students, with only a handful of exceptions, Dr. Kornbluth said. The loss adds up to about 500 students, which she said would hurt undergraduates who will have fewer research mentors.

“It’s a loss for the nation,” she said. “When you shrink the pipeline of basic discovery research, you choke off the flow of future solutions, innovations and cures — and you shrink the supply of future scientists.”

The California Institute of Technology, another of the nation’s premier research institutions, said it would reduce its new graduate students by 40 percent in fall 2026 across the board. The reason was not specific cuts, but a lack of funding certainty, David Chan, the university’s dean of graduate studies, said in an interview.

“The hope is to go back to historic numbers, but we don’t know when that will happen,” he said, adding that decisions for fall 2027 have not been made yet “given how quickly things are changing.”

Jessica Werk, the astronomy department chair at the University of Washington, said the institution was facing a mix of state budget cuts and worries about the Trump administration’s proposals.

“There’s all these external factors that are just kind of pressing in on us,” said Dr. Werk, whose department typically hires about four new doctoral students a year from an application pool of about 400.

She decided not to take any new doctoral students in fall 2026 — the first time at least since 2016, when she joined the department. She said the department planned to hire again in fall 2027.

The data from the A.A.U. institutions also show a decline in applications to Ph.D. programs — a gauge of demand — driven by international students. While applications from domestic students increased by 3 percent, applications from international students fell by 21 percent.

The Trump administration has taken a series of actions that experts say have dampened the desire of international students to study in the United States, including delaying visa interviews, expanding a ban on travel from certain countries and screening the social media posts of applicants for speech deemed anti-American.

The decline in applications from international students, Mr. Smith said, comes as other countries like China are making a strong recruitment push “to take advantage of the United States’ unforced errors.”