Quantum-Classical Proof Systems

• Quantum-classical proof systems are hybrid protocols that leverage both quantum and classical components to enable robust verification of quantum computations.
• They establish complexity separations and underpin privacy-preserving tools such as zero-knowledge proofs and proofs of quantum knowledge.
• Advanced techniques in error management, amplification, and simulation empower classical verifiers to reliably verify quantum phenomena even under adversarial conditions.

Quantum-classical proof systems are interactive or non-interactive protocols in which a prover and verifier may combine quantum resources with classical communication or proofs. These systems have emerged as central objects in quantum complexity theory, quantum cryptography, complexity-theoretic separations, and the paper of robust delegation and verification of quantum computation. The spectrum of models encompasses quantum verifiers with classical witnesses (QCMA), quantum interactive proof systems with unbounded and bounded error, quantum-classical PCPs, zero-knowledge, proofs of quantum knowledge, and the classical verification of quantum phenomena. This article surveys the foundational concepts, structural results, key methodologies, separations, and applications of quantum-classical proof systems as revealed in contemporary literature.

1. Core Models of Quantum-Classical Proof and Verification

Quantum-classical proof systems can be divided according to the quantum or classical nature of (a) the verifier, (b) the proof/witness, and (c) the interaction channel.

• QCMA: The witness is classical, but the verifier is quantum; Merlin sends a classical string and Arthur is a quantum polynomial-time verifier (Jordan et al., 2011). QCMA is contained in QMA, where the witness may be quantum.
• Quantum Interactive Proofs (QIP): Both the prover and verifier may exchange quantum messages; completeness-soundness gap, number of messages, error bounds, and public-coin structure yield a rich complexity landscape (Ito et al., 2010, Vidick et al., 2016).
• Arthur-Merlin Models with Restricted Verifiers: When the verifier is modeled as a two-way quantum finite automaton with constant-size quantum registers (2QCFA) and classical control, the model combines quantum succinctness with classical procedural logic, enabling verification of languages in PSPACE, i.e., QIP ⊆ AM(2QCFA) (Zheng et al., 2013, Yakaryılmaz, 28 Aug 2025).
• Quantum-Classical PCPs (QCPCP): The verifier is quantum, the proof is classical, and queries to the proof may be performed in superposition (quantum queries). Constant-query QCPCPs can be simulated by protocols with three classical queries and constant completeness-soundness gap (Buhrman et al., 1 Nov 2024).
• Proofs of Quantum Knowledge with Classical Verifiers: A classical verifier interacts with a quantum prover (or multiple quantum provers) and is convinced the prover has access to a quantum witness (possibly a quantum ground state), with the security certified via extractors with black-box quantum access (Vidick et al., 2020, Broadbent et al., 17 Mar 2025).
• Classical Verification of Quantum Computation and Learning: Through protocols such as Mahadev’s measurement protocol and its refinements (Vidick et al., 2019, Caro et al., 2023), classical verifiers can validate the outcomes of quantum computations or learning tasks by interacting with quantum provers and leveraging cryptographic primitives (e.g., LWE-based trapdoor claw-free functions).

The landscape thus encompasses both classical-witness/quantum-verifier (“QCMA-type”), quantum-witness/quantum-verifier (“QMA-type”), and protocols in which classical verifiers extract quantum phenomena through interaction and cryptography.

2. Amplification, Parallelization, and Error Bounds

Critical advances in quantum-classical proof systems have been centered around the manipulation of error parameters, interaction complexity, and witness structure.

• Error Tolerance and Bounded Error: The acceptance gap (completeness - soundness) plays a decisive role in the computational power of interactive quantum proof systems. In (Ito et al., 2010), it is shown that with only a double-exponentially small completeness-soundness gap (i.e., error difference Δ = 2^{–2^{poly(n)}}), quantum interactive proofs suffice to recognize EXP, far surpassing the PSPACE upper bound for classical interactive proofs with similar error.
• Parallelization and SDP Characterization: Quantum interactive proofs can be parallelized to yield three-message (turn) protocols without loss of power, using semidefinite programming (SDP) formulations to encode the optimal strategy for the prover (Vidick et al., 2016). This is fundamental for the PSPACE containment QIP = PSPACE.
• Perfect Completeness in QCMA: QCMA proof systems can be amplified to perfect completeness (i.e., acceptance probability 1 for all yes-instances) under any gate set allowing exact implementation of Hadamard and classical reversible gates; this involves additive amplitude recalibration followed by quantum rewinding (Jordan et al., 2011).
• Promise Gap Amplification in QCPCPs: Any constant-query QCPCP (with even inverse polynomial c–s gap) can be reduced to a protocol with constant gap and only three (classical) queries via the polynomial method and the BQ-operator, thereby placing the class inside BQ•NP (Buhrman et al., 1 Nov 2024).

This structural malleability has major implications; for instance, relaxing error allows exponential jumps in power (PSPACE vs. EXP) and, in particular, bounded-error is essential for equivalence results (QIP = PSPACE). Similarly, query and round reduction arguments are necessary for robust PCP constructions.

3. Separations, Oracle Results, and Robustness

Quantum-classical proof systems are fertile ground for the demonstration of complexity-theoretic separations and collapse results.

• QCMA vs. QMA (Oracle Separations): Oracle constructions show that QCMA verifiers (quantum verifier, classical witness) cannot in general verify certain subset-structured properties efficiently, whereas QMA (quantum witness) can. Notably, in-place permutation oracles yield an exponential separation in query complexity (Fefferman et al., 2015).
• QCPCP and Uniqueness: Constraining quantum-classical PCP systems to unique proofs (UniqueQCPCP) does not alter their computational power, i.e., UniqueQCPCP = QCPCP, under BQ-operator and randomized reductions (Anand et al., 24 Jun 2025); this aligns with classical intuition from the Valiant–Vazirani theorem and supports the robustness of proof models under uniqueness constraints.
• A Non-uniform Quantum Karp–Lipton Theorem: If QMA ⊆ BQP/qpoly, then the Quantum Polynomial Hierarchy (QPH) collapses to its second level with quantum advice (QΣ₂/qpoly), paralleling and extending the classical Karp–Lipton theorem (Anand et al., 24 Jun 2025).
• Collapse in Consistent Quantum Polynomial Hierarchy: CQPH, requiring proofs in successive rounds to be consistent extensions, collapses unconditionally to CQΣ₂ via convexity and minimax arguments, in contrast to entanglement-driven collapses in the QEPH setting (Anand et al., 24 Jun 2025).
• Limitations of Quantum-Classical PCPs for QCMA: There is strong evidence (supplemented by oracular arguments) that no constant-query quantum-classical PCP exists for QCMA, as this would imply QCMA ⊆ BQ·NP, an inclusion believed to be false (Buhrman et al., 1 Nov 2024).

These separation and collapse results clarify the boundaries of quantum-classical proof power, the roles of key structural constraints, and the interplay between quantumness, advice, and verification.

4. Zero-Knowledge, Proofs of Quantum Knowledge, and Simulation Techniques

Quantum-classical proof systems drive the development of advanced tools for privacy-preserving verification and knowledge extraction in the quantum regime.

• Quantum Zero-Knowledge for QMA: Under cryptographic assumptions (e.g., unconditionally binding and quantum computationally concealing commitment schemes), every problem in QMA has a quantum zero-knowledge proof system (Broadbent et al., 2016). The protocol leverages strong encoding (concatenated code with trap qubits and one-time pad), commitment, coin-flipping, and classical NP zero-knowledge proofs to ensure that nothing about the quantum witness is revealed aside from the validity of the claim.
• Classical-Verifier Zero-Knowledge for Quantum Computation: Composition of the Mahadev measurement protocol with zero-knowledge encodings enables the first classical-verifier zero-knowledge arguments for QMA, making it possible for a purely classical client to verify quantum computations (e.g., ground state energy of local Hamiltonians) without learning the witness (Vidick et al., 2019).
• Proofs of Quantum Knowledge (PoQK): In both single- and multi-prover settings, a classical verifier can be convinced that a prover possesses a quantum state (as a witness), with the PoK property certified by the existence of a quantum extractor (with black-box access) that reconstructs the state if the protocol accepts with high probability. Multi-prover protocols leverage self-testing and swap gadgets to extract witnesses from strategies passing the local Hamiltonian energy test (Vidick et al., 2020, Broadbent et al., 17 Mar 2025).
• Quantum Rewinding and Simulation Techniques: Proofs exploit forms of “quantum rewinding” to simulate or extract interactions, using either exact or approximate lemmas (e.g., for zero-knowledge or knowledge-extraction), often under conditions such as constant or only slightly input-dependent success probability (Vidick et al., 2016).

These advances drive a privacy- and knowledge-centric view of quantum verification, ensuring composability and security in quantum cryptography and delegated computation.

5. Applications in Learning, Delegation, and Space-Bounded Verification

Quantum-classical proof systems have been adapted to novel application domains:

• Classical Verification of Delegated Quantum Learning: Classical verifiers, with only random or statistical query access, can efficiently verify agnostic learning (e.g., parities or Fourier-sparse Boolean functions) by interacting with quantum provers in a mixture-of-superpositions quantum data model. Classical verification protocols (e.g., requesting heavy Fourier coefficients) suffice to establish agnostic generalization guarantees—even in cases where classical learning alone would be infeasible (Caro et al., 2023).
• Sublogarithmic-Space Quantum Verification: Quantum Merlin–Arthur proof systems with the verifier’s space bounded to o(log n) (but ω(1)) are shown to verify nontrivial padded QMA-complete problems using only subpolynomial-length quantum proofs. No polynomial-time classical or stand-alone quantum algorithm can verify these problems under standard complexity assumptions, demonstrating unconditional quantum advantage in this low-space regime (Say, 13 May 2025).
• Simulation and Inter-simulability via Classical Circuits: Finite-dimensional quantum systems are equivalent under local (gauge) group transformations, allowing simulatability of one quantum system by another, and even simulation of quantum dynamics through classical network models (e.g., LC circuits) constructed by mapping the Schrödinger equation into real differential forms (Caruso, 2021). This supports efficient benchmarking and experimental validation strategies for proof systems.

These instances highlight the flexibility of quantum-classical proof architectures for addressing real-world verification and delegation challenges, from machine learning to quantum simulation.

6. Security, Hardness, and Post-Quantum Considerations

Quantum attacks reshape foundational notions in classical proof systems and cryptographic protocols:

• Hardness of Quantum Rewinding and Security Breaks: Rewinding-based security proofs, core to classical sigma-protocols, the Fiat–Shamir heuristic, and proof-of-knowledge constructions, often fail in the quantum setting due to the impossibility of “resetting” quantum states. The “pick-one trick” demonstrates that quantum adversaries can produce a single valid transcript but preclude extraction of a second, invalidating classical knowledge arguments (Ambainis et al., 2014).
• Quantum Attacks on Binding Commitments: Even computationally binding commitments may be “opened” to any value by a quantum adversary (but only to one value per commitment), making standard binding definitions problematic (Ambainis et al., 2014).
• Implications for Post-Quantum Cryptography: These challenges underscore the need for quantum-aware transformations (e.g., strict soundness, non-rewinding simulation) in cryptographic protocol design and the reevaluation of proof techniques and definitions to retain meaning against quantum attacks.

This security dimension reflects a deeper structural divergence of proof-of-knowledge and zero-knowledge protocols between classical and quantum settings and defines the direction for robust, post-quantum protocol engineering.

7. Outlook and Open Directions

Recent progress has mapped key boundaries of quantum-classical proof power, demonstrated quantum advantage in restricted models, and clarified where amplification, collapse, and separations occur. Current open directions include:

• Determining whether round and query reduction techniques for QCPCPs and quantum-classical interactive proofs extend beyond constant regimes, especially in the context of QCMA (Buhrman et al., 1 Nov 2024).
• Exploring whether classical-verifier zero-knowledge protocols can be further streamlined to non-interactive or fewer-round structures in practical cryptographic settings (Vidick et al., 2019).
• Deepening the integration of self-testing with knowledge extraction for robust proofs of quantumness, delegation, and verification in multi-prover scenarios (Broadbent et al., 17 Mar 2025).
• Refining simulation techniques that retain quantitative tightness for hardness and security against arbitrary quantum attacks (Ambainis et al., 2014).
• Classifying the complexity and verification power of hybrid models where the verifier (or proof) accesses only sublinear quantum resources, or where classical simulation bridges quantum behavior (Say, 13 May 2025, Caruso, 2021).

Quantum-classical proof systems thus remain a vibrant intersection of quantum theory, complexity, cryptography, and practical verification with continued evolution expected as new theoretical tools, cryptographic constraints, and quantum technologies emerge.