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arXiv:2610.00847

Beyond IP = PSPACE and QIP = PSPACE: Interactive Proofs in Arbitrary Physical Theories

Kishor Bharti

quant-phcs.CCSubmitted Oct 1, 2026• Announced Oct 2, 2026• 0 citations

Abstract

The equalities IP = PSPACE and QIP = PSPACE, the latter achievable with three messages, raise a basic question: how much of an interactive proof's power comes from the underlying physical theory? We study interactive proofs in general probabilistic theories, which include classical and quantum theory. The answer depends on what the prover and verifier exchange and how the theory specifies efficient operations. When they exchange only classical messages, protocols in every theory satisfying our standard assumptions decide exactly PSPACE. For protocols with a quantum verifier and quantum messages, allowing a prover to use any theory containing quantum theory does not increase the maximum acceptance probability. Thus, the three-message PSPACE result remains valid against such provers. When messages may be arbitrary systems, the interactive-proof class can strictly exceed PSPACE.

Comments: 36 pages

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The equalities IP = PSPACE and QIP = PSPACE, the latter achievable with three messages, raise a basic question: how much of an interactive proof's power comes from the underlying physical theory? We study interactive proofs in general probabilistic theories, which include classical and quantum theory. The answer depends on what the prover and verifier exchange and how the theory specifies efficient operations. When they exchange only classical messages, protocols in every theory satisfying our standard assumptions decide exactly PSPACE. For protocols with a quantum verifier and quantum messages, allowing a prover to use any theory containing quantum theory does not increase the maximum acceptance probability. Thus, the three-message PSPACE result remains valid against such provers. When messages may be arbitrary systems, the interactive-proof class can strictly exceed PSPACE.

We introduce quantum interactive proof systems, which are interactive proof systems in which the prover and verifier may perform quantum computations and exchange quantum messages. It is proved that every language in PSPACE has a quantum interactive proof system that requires a total of only three messages to be sent between the prover and verifier and has exponentially small (one-sided) probability of error. It follows that quantum interactive proof systems are strictly more powerful than classical interactive proof systems in the constant-round case unless the polynomial time hierarchy collapses to the second level.

In this paper we consider quantum interactive proof systems, i.e., interactive proof systems in which the prover and verifier may perform quantum computations and exchange quantum messages. It is proved that every language in PSPACE has a quantum interactive proof system that requires only two rounds of communication between the prover and verifier, while having exponentially small (one-sided) probability of error. It follows that quantum interactive proof systems are strictly more powerful than classical interactive proof systems in the constant-round case unless the polynomial time hierarchy collapses to the second level.

We introduce a new variant of quantum multiprover interactive proofs (QMIP) where the provers and the verifier are quantum. The verifier can exchange quantum messages with the provers. The provers cannot communicate quantumly between themselves and do not share entanglement, but are unlimited in the classical communication between them, even after receiving messages from the verifier. We show that any language in nondeterministic exponential time (NEXP) can be recognized in this model efficiently, with just two provers and two rounds of communication, and with a constant completeness/soundness gap. This is in contrast to the result of [R. Jain et al., Comm. ACM, 53 (2010), pp. 102--109], which shows that QIP = PSPACE, or equivalently that the set of languages that can be recognized by a quantum verifier communicating with a single quantum prover is equal to PSPACE. To analyze the cheating power of the provers, we give them more power and allow them to perform any separable operation. We then show a unique two-phase protocol in which the provers first commit to a superposition of correct answers to all possible questions, and then in the second phase the verifier opens up the committed answer and checks for correctness and consistency.

We explore quantum-inspired interactive proof systems where the prover is limited. Namely, we improve on a result by \cite{AG17} showing a quantum-inspired interactive protocol ($\IP$) for

PreciseBQP

where the prover is only assumed to be a $\PreciseBQP$ machine, and show that the result can be strengthened to show an $\IP$ for $\NP^{\PP}$ with a prover which is only assumed to be an $\NP^{\PP}$ machine - which was not known before. We also show how the protocol can be used to directly verify $\QMA$ computations, thus connecting the sum-check protocol by \cite{AAV13} with the result of \cite{AG17,LFKN90}. Our results shed light on a quantum-inspired proof for $\IP=\PSPACE$, as $\PreciseQMA$ captures the full $\PSPACE$ power.