Determine the relationship between QMA(2) and QMA

Determine whether the complexity class QMA(2), consisting of quantum verification with two unentangled proofs, is strictly more powerful than QMA in the unrelativized setting, or characterize their precise relationship beyond the known containments QMA ⊆ QMA(2) ⊆ NEXP.

Background

The paper studies the computational power gained by allowing a verifier to receive two quantum proofs promised to be unentangled. Before this work, the general relationship between QMA(2) and QMA was largely unresolved, with essentially only the containments QMA ⊆ QMA(2) ⊆ NEXP known.

The paper resolves a relativized version by constructing a unitary oracle relative to which QMA and QMA(2) differ. This does not settle the corresponding unrelativized complexity-class relationship, so the broader problem remains open.

References

Whether this model is more powerful than $\mathsf{QMA}$ remains open .

— PureSuperQMA(exp) = BellPureSymQMA(poly) = QMA via Dimension-Free Bosonic Argmax  (2609.11854 - Gay et al., 10 Sep 2026) in Section 1, subsection “Purity and quantum proof systems”; reiterated in subsection “Our results”

Understanding the power of $QMA(2)$ is a major open problem in quantum complexity theory and is closely connected to fundamental questions about entanglement, separability, and optimization; see the recent survey of Jeronimo, Wu, and Leigh.

— A quantum oracle separation between QMA(2) and QMA  (2609.02865 - Bostanci et al., 2 Sep 2026) in Section 1, Introduction

Could the techniques in this paper be used to exhibit a property which has exponential copy-complexity, yet is efficiently testable given $k \geq 2$ copies of a proof?

— A quantum oracle separation between QMA(2) and QMA  (2609.02865 - Bostanci et al., 2 Sep 2026) in Section 6, Conclusion, “Future directions,” item 3

Thus, whether $QMA\overset{?}{=}$ remains a difficult open question.

— Semidefinite extension complexity of the separable set, with applications to approximate disentanglers  (2609.09033 - Gharibian et al., 8 Sep 2026) in Section 1, Introduction, first paragraph