Resolve the Quantum PCP (QPCP) conjecture

Resolve the quantum PCP conjecture by proving or refuting whether every QMA language admits a quantum PCP with constant query complexity and polynomial overhead, thereby establishing (or disproving) the quantum analogue of the classical PCP theorem.

Background

The paper highlights a conceptual connection between zero‑knowledge PCPs and quantum PCPs via quantum codes: good code distance corresponds to hiding information (akin to zero knowledge). While there is evidence for good quantum LTCs, local decodability is blocked by no‑cloning, and QPCP remains a central open problem in quantum complexity.

By drawing parallels, the authors suggest that studying perfect zero‑knowledge PCPs could shed light on QPCP, but they do not provide a specific formulation beyond pointing to the conjecture’s status.

References

Finally, we highlight a connection between zero-knowledge PCPs and one of the most imporant open problems in quantum complexity theory: the quantum PCP (QPCP) conjecture.

— A Zero-Knowledge PCP Theorem  (2411.07972 - Gur et al., 2024) in Subsection “Open problems”, Section 1 (Introduction), paragraph “The quantum PCP conjecture.”

Quantum locally testable codes are motivated in part by the quantum PCP conjecture, which concerns the hardness of approximating the ground energy of a local Hamiltonian to constant additive accuracy after normalization .

— Asymptotically Good Quantum Locally Testable Codes  (2609.20780 - Gay et al., 17 Sep 2026) in Introduction, paragraph beginning “Quantum locally testable codes are motivated in part...”

However, obtaining a succinct argument for $\QMA$ via their transformation requires a QPCP for $\QMA$ with polynomial length and small query complexity, whose existence is a weak form of the QPCP conjecture and remains a major open problem in quantum complexity theory.

— Succinct Arguments for QMA in the Quantum Random Oracle Model  (2609.37429 - Chiesa et al., 29 Sep 2026) in Section 1, Introduction