We show the following hold, unconditionally unless otherwise stated, relative to a random oracle with probability 1: •There are NP search problems solvable by BQP machines but not BPP machines.•There exist functions that are one-way, and even collision resistant, against classical adversaries but are easily inverted quantumly. Similar separations hold for digital signatures and CPA-secure public key encryption (the latter requiring the assumption of a classically CPA-secure encryption scheme). Interestingly, the separation does not necessarily extend to the case of other cryptographic objects such as PRGs.•There are unconditional publicly verifiable proofs of quantumness with the minimal rounds of interaction: for uniform adversaries, the proofs are non-interactive, whereas for non-uniform adversaries the proofs are two message public coin.•Our results do not appear to contradict the Aaronson-Ambanis conjecture. Assuming this conjecture, there exist publicly verifiable certifiable randomness, again with the minimal rounds of interaction.By replacing the random oracle with a concrete cryptographic hash function such as SHA2, we obtain plausible Minicrypt instantiations of the above results. Previous analogous results all required substantial structure, either in terms of highly structured oracles and/or algebraic assumptions in Cryptomania and beyond.
We obtain a certified randomness protocol in the quantum random oracle model. The protocol is non-interactive and publicly verifiable with a classical verifier, and is based on Yamakawa and Zhandry's proof of quantumness [JACM'24]. We prove unconditional security of this protocol against adversaries making subexponentially-many adaptive quantum queries to the random oracle.
Prior work on certified randomness relative to a random oracle additionally assumed the Aaronson--Ambainis conjecture or proved security only against low query-depth adversaries.
We show the following hold, unconditionally unless otherwise stated, relative to a random oracle: — There are NP search problems solvable by quantum polynomial-time (QPT) machines but not classical probabilistic polynomial-time (PPT) machines. — There exist functions that are one-way, and even collision resistant, against classical adversaries but are easily inverted quantumly. Similar counterexamples exist for digital signatures and CPA-secure public key encryption (the latter requiring the assumption of a classically CPA-secure encryption scheme). Interestingly, the counterexample does not necessarily extend to the case of other cryptographic objects such as PRGs. — There are unconditional publicly verifiable proofs of quantumness with the minimal rounds of interaction: for uniform adversaries, the proofs are non-interactive, whereas for non-uniform adversaries the proofs are two message public coin. — Our results do not appear to contradict the Aaronson-Ambanis conjecture. Assuming this conjecture, there exist publicly verifiable certifiable randomness, again with the minimal rounds of interaction. By replacing the random oracle with a concrete cryptographic hash function such as SHA2, we obtain plausible Minicrypt instantiations of the above results. Previous analogous results all required substantial structure, either in terms of highly structured oracles and/or algebraic assumptions in Cryptomania and beyond.
In a recent breakthrough, Yamakawa and Zhandry (J. ACM 2024) constructed a proof of quantumness in the quantum random oracle model (QROM) in which the quantum prover samples a codeword preimage of a publicly computable function H. They conjectured that given any H, a successful prover must sample their preimage from a high-entropy distribution over possible answers. If true, this would give a certifiable randomness protocol in the quantum random oracle model. As partial evidence for their conjecture, Yamakawa and Zhandry proved the security of their certifiable randomness protocol assuming the Aaronson-Ambainis conjecture.
We prove the security of the certifiable randomness protocol of Yamakawa-Zhandry unconditionally, without relying on the unproven Aaronson-Ambainis conjecture, against low query-depth quantum adversaries: specifically, adversaries that make up to o(\log λ) adaptive quantum queries to the random oracle.
We provide the first evidence relative to a classical oracle that pseudorandom unitaries (PRUs) can exist without quantum-computable one-way functions.
To obtain this result, we first prove that five independent random diagonal phase layers interleaved with Hadamard transforms
U=F5HF4HF3HF2HF1 form a strong PRU with security under adaptive, controlled access to the unitary, its inverse, transpose, and complex conjugate.
Our main result is that, when the phase layers are implemented using a classical random oracle O, the construction remains secure against uniform BQP adversaries with coherent access to suitable Boolean completions of classical-input $QMA^{PH}^{O}}$ decision problems. Such an adversary can invert (even quantum-computable) one-way functions and solve efficiently verifiable one-way puzzles defined relative to O. Our result therefore provides classical-oracle evidence that PRUs do not imply quantum-computable one-way functions, in the form of a black-box separation.
We substantially strengthen the classical-oracle result of Kretschmer, Qian, and Tal (STOC 2025) by obtaining PRUs secure against these QMA-aided adversaries. Our construction also consists of a simple, efficient circuit that queries a random function, suggesting a route to concrete implementation via the random-oracle heuristic.
Whether some problems require quantum proofs has been a central question in quantum complexity (Aharonov and Naveh, 2002; Aaronson and Kuperberg, CCC 2007). Recently, breakthrough work of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC 2026), followed by a simpler separation due to Bostanci, Huang, and Vaikuntanathan (FOCS 2026), established a classical oracle separation between
QMA and
QCMA. However, while they are not in
QCMA, the problems used in both separations still lie in
AM: they admit a two-message public-coin proof system with a classical verifier. In this work, we ask whether some problems truly require quantum proofs. More formally, we consider the complexity class
QCIP, introduced by Buhrman, Le Gall, and Weggemans (2024), where an efficient quantum verifier interacts with an unbounded prover over a classical channel for an arbitrary polynomial number of rounds. We construct a classical oracle
O such that
QMAO⊆QCIPO, thus showing that some languages indeed require quantum proofs, with no classical replacements. Since
QCMA=QCIP[1], this strengthens the earlier
QMA--
QCMA separations, which now follow as a special case of our result. As a technical contribution, we extend to the complexity theory setting the techniques developed by Cakan, Goyal, and Shmueli (CRYPTO 2026) in the context of cryptography for analyzing classically interacting quantum machines. We believe this may be of independent interest.
In classical cryptography, one-way functions are necessary for most of cryptography. In the quantum world, however, Kretschmer at {TQC'21} and subsequent works have demonstrated that quantum versions of various primitives in minicrypt, such as signatures and private-key encryption, do not require quantum-computable one-way functions (\textsf{qOWF}s). However, the question of the necessity of \textsf{qOWF}s for primitives in Cryptomania like public-key encryption with classical public keys and ciphertexts (\textsf{qPKE}) has remained open.
We provide a comprehensive picture of the relationship between \textsf{qPKE} and \textsf{qOWF}. First, our main result is a black-box separation between \textsf{qOWF}s and \textsf{qPKE}. In particular, we show that relative to a unitary, real, and self-inverse oracle, there exists a \textsf{qPKE} scheme (with classical keys and ciphertexts), but \textsf{qOWF}s do not exist. Further, our oracles admit a point-function obfuscation construction, thus ruling out black-box construction of \textsf{qOWF} from point function obfuscations.
On the positive side, we construct \textsf{qPKE} from quantum indistinguishability obfuscation for quantum circuits with classical obfuscated descriptions and efficiently verifiable one-way puzzles, which is known to be a potentially weaker assumption than \textsf{qOWF}s, due to an existing black-box separation. Our construction is the quantum analog of the classical construction of \textsf{PKE} from one-way functions and indistinguishability obfuscation in the quantum computation with classical communication (QCCC) setting.