Multiple unentangled quantum proofs

The verifier may jointly process the two witnesses, but the honest and cheating witnesses in QMA(2) must be product across the two prover registers. The product-state optimization underlying this model connects it to separability and tensor optimization.

What is known: Harrow–Montanaro show that polynomially many unentangled witnesses collapse to two, with the usual constant completeness–soundness gap. Very short proofs require attention to that gap: logarithmic-size protocols with inverse-polynomial gap are not the same claim as the $\widetilde O(\sqrt n)$-size, constant-gap two-witness protocols for NP problems. The general upper bound is $\mathsf{QMA}(2)\subseteq\mathsf{NEXP}$; the intermediate target $\mathsf{QMA}(2)\subseteq\mathsf{EXP}$ is also open.

July 2026 preprint: Beckey, Jeronimo and Wu determine the exact worst-case product-test acceptance curve over all product overlaps and finite local dimensions. This settles the low-overlap analysis and strengthens one-shot soundness of a key reduction, without settling the class comparison.

September 2026 preprint: Bostanci et al. construct a quantum oracle separating QMA(2) from QMA. They also prove the geometric no-disentanglers conjecture at constant errors $\varepsilon+\delta<1$: an approximate disentangler needs exponentially many input qubits in the number of output qubits. The class separation is relativized; the geometric theorem is not. Neither proves an unrelativized separation, and a classical-oracle separation remains open. Related: A3, E3.

Source disclosures

Product-test authors report ChatGPT 5.5 Pro assistance with analysis, ChatGPT/Claude assistance with writing and exploratory Codex-supported Lean work; they report human verification and responsibility for the manuscript. Source

The September authors report that ChatGPT 5.6 Sol suggested the central proof idea after prompts about She–Yuen, and that they verified, simplified and developed it. This records their disclosure, not an independent provenance audit. Source

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

preprint: Exact product-test acceptance curve, including the previously open low-overlap regime. Evidence

preprint: Quantum-oracle QMA(2)/QMA separation and an unrelativized no-disentanglers theorem at constant errors with epsilon + delta < 1. Evidence