Since the introduction of the complexity class QMA as a quantum-verifier analogue of NP (Kitaev, 1997), many have wondered whether quantum proofs are necessary or classical proofs suffice - that is, whether QMA = QCMA or QCMA != QMA (Aharonov and Naveh, 2002; Aaronson and Kuperberg, CCC '07). This longstanding question was recently answered by works of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC '26) and Bostanci, Huang, and Vaikuntanathan (FOCS '26), which showed that quantum proofs are more powerful than classical ones in the classical-oracle setting.
However, it remains unclear what exactly makes quantum proofs more powerful than classical ones. In the information-theoretic setting, a family of quantum states is not classicalizable if and only if it is unclonable. Indeed, these recent works also explicitly highlight the unclonability of their quantum proofs as a key mechanism behind their separations, and their arguments crucially rely on this property. This raises the question of whether unclonability is necessary for quantum proofs to be more powerful than classical ones.
In this work, we show that even clonable quantum proofs can be more powerful than classical ones relative to a classical oracle by constructing a classical oracle O such that QCMA^O != ClonableQMA^O. This resolves the open question of Nehoran and Zhandry (ITCS '24), who established the analogous separation relative to a quantum oracle. We also show a classical-oracle separation between BQP/clonableqpoly and BQP/poly, and give applications of our results to quantum cryptography.
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.
Yes, we show that they are. We initiate the study of languages that necessarily need uncloneable quantum proofs and advice. We define strictly uncloneable versions of the classes QMA, BQP/qpoly and FEQP/qpoly (which is the class of relational problems solvable exactly with polynomial-sized quantum advice). Strictly uncloneable QMA is defined to be the class of languages in QMA that only have uncloneable proofs, i.e., given any family of candidate proof states, a polynomial-time cloning algorithm cannot act on it to produce states that are jointly usable by
k separate polynomial-time verifiers, for arbitrary polynomial
k. This is a stronger notion of uncloneable proofs and advice than those considered in previous works, which only required the existence of a single family of proof or advice states that are uncloneable. We show that in the quantum oracle model, there exist languages in strictly uncloneable QMA and strictly uncloneable BQP/qpoly. The language in strictly uncloneable QMA also gives a quantum oracle separation between QMA and the class cloneableQMA introduced by Nehoran and Zhandry (2024). We also show without using any oracles that the language, used by Aaronson, Buhrman and Kretschmer (2024) to separate FEQP/qpoly and FBQP/poly, is in strictly uncloneable FEQP/qpoly.
It is a long-standing open question to construct a classical oracle relative to which BQP/qpoly
= BQP/poly or QMA
= QCMA. In this paper, we construct classically-accessible classical oracles relative to which BQP/qpoly
= BQP/poly and QMA
= QCMA. Here, classically-accessible classical oracles are oracles that can be accessed only classically even for quantum algorithms. Based on a similar technique, we also show an alternative proof for the separation of QMA and QCMA relative to a distributional quantumly-accessible classical oracle, which was recently shown by Natarajan and Nirkhe.
We study whether a quantum verifier can find a hidden quantum state by exchanging classical messages with an unbounded prover who knows the state. The verifier has access to a reflection oracle about the target. For a Haar-random
n-qubit target, we prove that achieving constant average fidelity with the target requires
(q+1)2(B+1)=Ω(2n), where
q is the number of verifier queries and
B is the total prover-to-verifier communication. The bound allows arbitrary classical interaction and unrestricted computation between queries. Our proof extends the lower bound of Aaronson and Kuperberg (ToC'07) from a single classical proof to an arbitrary classical interaction.
As a result, we obtain a quantum-oracle separation of QMA from QCIP. In particular, we give a unitary oracle
U such that there exists a language
L∈QMAU but
L∈/QCIPU. Our separation holds even for UniqueQMA, the restriction of QMA to verification with a unique quantum witness, which is known to be strictly weaker than QMA in the quantum black-box setting.
We further give an oracle construction for a notion known as QMA--QCIP[2] gap problem. This gives average case hardness for a QMA language under classical access to a QCIP[2] decision oracle and is used in establishing negative results on witness encryption and quantum indistinguishability obfuscation.
We construct a classical oracle proving that, in a relativized setting, the set of languages decidable by an efficient quantum verifier with a quantum witness (QMA) is strictly bigger than those decidable with access only to a classical witness (QCMA). The separating classical oracle we construct is for a decision problem we coin spectral Forrelation -- the oracle describes two subsets of the boolean hypercube, and the computational task is to decide if there exists a quantum state whose standard basis measurement distribution is well supported on one subset while its Fourier basis measurement distribution is well supported on the other subset. This is equivalent to estimating the spectral norm of a "Forrelation" matrix between two sets that are accessible through membership queries.
Our lower bound derives from a simple observation that a query algorithm with a classical witness can be run multiple times to generate many samples from a distribution, while a quantum witness is a "use once" object. This observation allows us to reduce proving a QCMA lower bound to proving a sampling hardness result which does not simultaneously prove a QMA lower bound. To prove said sampling hardness result for QCMA, we observe that quantum access to the oracle can be compressed by expressing the problem in terms of bosons -- a novel "second quantization" perspective on compressed oracle techniques, which may be of independent interest. Using this compressed perspective on the sampling problem, we prove the sampling hardness result, completing the proof.