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<\/p>\n
Without any extra information, this problem is hard to solve even for a quantum computer. But given an appropriate quantum state, a quantum computer can easily confirm that it\u2019s consistent with both shadows. In other words, that state is a valid quantum proof.<\/p>\n
Now imagine you\u2019re instead given a written procedure for how to generate a quantum state consistent with both shadows. That procedure would count as a classical proof for the spectral forrelation problem: To check that it\u2019s valid, you\u2019d first run the procedure on your quantum computer, then compare the resulting state to the two shadows. It doesn\u2019t sound like a traditional mathematical proof, but it would still be a concise written document rather than a quantum state that\u2019s too complex to write down.<\/p>\n
Zhandry needed to show that classical proofs can\u2019t exist. He sought to do so with a strategy called a proof by contradiction. First, he\u2019d assume the opposite of what he wanted to prove: that a classical proof for the spectral forrelation problem is possible. Then he\u2019d need to show that this assumption would eventually lead to a contradiction.<\/p>\n
That contradiction, he suspected, would come from a property of classical proofs that we usually take for granted: It\u2019s possible to read a proof more than once.<\/p>\n
Chasing Shadows<\/p>\n
Outside of spy movies, documents rarely self-destruct after they\u2019re read \u2014 and fortunately for mathematicians, proofs are no exception. But in the quantum world, things are different: Measuring a quantum state can irreversibly disturb it, altering the results of any subsequent measurements. This kind of measurement disturbance plays a central role<\/a> in many quantum cryptography schemes, but researchers hadn\u2019t exploited it in previous attempts to distinguish between quantum and classical proofs.<\/p>\n The final step turned out to have a fatal flaw. Coming so tantalizingly close made them all the more determined to succeed.<\/p>\n Coming from a background in cryptography, however, Zhandry saw that measurement disturbance could be relevant. A quantum proof for the spectral forrelation problem is a quantum state that\u2019s vulnerable to measurement disturbance. A hypothetical classical proof, on the other hand, would be a written document, such as a procedure for generating a valid quantum state. Anyone could run the procedure repeatedly to churn out fresh copies of that state.<\/p>\n Zhandry wanted to explore the implications of this reusability, because he suspected it was too good to be true.<\/p>\n He quickly showed<\/a> that if a classical proof for the spectral forrelation problem existed, anyone with a copy of the proof could use it repeatedly to accomplish a seemingly difficult task: guessing the shapes of shadows given only partial information. Only one step remained. If Zhandry could separately prove that this guessing task was not just hard but so hard that even a classical proof couldn\u2019t help, he would have a contradiction. That would mean his starting assumption, that classical proofs were possible, had to be false.<\/p>\n Zhandry couldn\u2019t figure out how to complete that last step alone, so at the end of 2024 he teamed up with John Bostanci<\/a>, now a researcher at the Simons Institute for the Theory of Computing in Berkeley, California, and Jonas Haferkamp<\/a>, a computer scientist now at Ruhr University Bochum in Germany. Soon the trio had what they thought was a finished proof \u2014 but the final step turned out to have a fatal flaw. Coming so tantalizingly close made them all the more determined to succeed.<\/p>\n \u201cThat kind of lit the fire under our butts,\u201d Bostanci said.<\/p>\n Nirkhe, who\u2019d been wrestling with the problem independently<\/a> for years, joined the team in early 2025 and suggested a way to tweak Zhandry\u2019s approach. They could use the same overall strategy, but almost every detail would have to change. Nirkhe\u2019s proposal kicked off a nine-month period full of overstuffed emails and travel back and forth between New York, Washington state, California, and Germany.<\/p>\n \u201cIt really dominated my year,\u201d Bostanci said. \u201cI basically didn\u2019t do much else.\u201d<\/p>\n The four researchers chipped away at the problem by drawing on ideas from other areas of physics and computer science, including quantum learning theory<\/a> and the math of quantum particles called bosons<\/a>. One crucial breakthrough came in the early fall while Bostanci was in the middle of a 20-mile run in New York City\u2019s Central Park, part of his training for the upcoming marathon.<\/p>\n After two more months of intense work, the team finally succeeded. They\u2019d reached a contradiction, which meant that their original assumption had to be wrong: A classical proof for the spectral forrelation problem was impossible. They posted their result online in mid-November, 10 days after Bostanci successfully finished his race.<\/p>\n Proof of Concept<\/p>\n Officially, the team proved, with one caveat, that two classes of computational problems are different. One class includes all problems with quantum proofs and is known as QMA. The other, called QCMA, includes problems with classical proofs that a quantum computer can check. (The unwieldy acronyms stand for quantum Merlin-Arthur and quantum-classical Merlin-Arthur, respectively, in reference to a fanciful thought experiment<\/a> featuring the two characters from medieval legend.)<\/p>\n The caveat is that the team\u2019s result is an \u201coracle separation\u201d between QMA and QCMA. This means it relies on certain assumptions that restrict the space of possibilities one needs to consider. But it\u2019s strong evidence that quantum proofs are more powerful than classical ones \u2014 precisely the kind of evidence that researchers have sought for 20 years.<\/p>\n Soon after the team posted their paper online, an MIT master\u2019s student named Andrew Huang<\/a> heard Bostanci give a talk about the result. He realized that one aspect of the team\u2019s proof could also play a role in an oracle separation based on a completely different computational problem<\/a>. Huang and his adviser, Vinod Vaikuntanathan<\/a>, teamed up with Bostanci and soon proved a second oracle separation<\/a> between QMA and QCMA. The newer result further bolsters the case that quantum proofs are inherently more powerful than classical ones.<\/p>\n The techniques used to prove these oracle separations could one day find applications in cryptography. But for many researchers, the allure of the \u201cQMA versus QCMA\u201d question doesn\u2019t come from any potential practical application. It offers a way to explore deep philosophical questions about quantum theory that have vexed physicists for over a century<\/a>.<\/p>\n \u201cMy real interest has always been, \u2018Why is quantum mechanics not classically describable?\u2019\u201d Nirkhe said. \u201cI think of computation as the yardstick, or the metric, with which we can understand this.\u201d<\/p>\n Editor\u2019s note: Scott Aaronson is a member of Quanta Magazine\u2019s advisory board.<\/p>\n ![\"\"](\"https:\/\/www.newsbeep.com\/ie\/wp-content\/uploads\/2026\/07\/additional-spot.webp\")
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