Effectiveness hypotheses and the boxworld interactive-proof class

Determine which hypothesis on message state spaces, expressed in the coordinates of the computable verifier gates, guarantees that the interactive-proof class equals PSPACE for a natural class of theories including theories not isomorphic to quantum or classical theory, and determine whether the interactive-proof class of boxworld equals PSPACE.

Background

The paper proves PSPACE characterizations for protocols with classical messages in arbitrary theories satisfying the standing assumptions and for quantum-verifier protocols when the theory contains ordinary quantum theory. For arbitrary message systems, however, the state-space geometry may encode computationally hard or noncomputable real numbers even when verifier gates have efficiently computable coordinates. The triangular theories constructed in the paper demonstrate that additional assumptions relating message state spaces to gate coordinates are necessary.

The authors note that boxworld has polyhedral message state spaces with rational vertices in the specified coordinates, so the obstruction exhibited by the triangular construction is not apparent there. Nevertheless, no PSPACE upper bound is established for boxworld because the paper does not obtain a finite, quantum-comb-like description of general boxworld prover strategies.

References

Which hypothesis on the message state spaces, formulated in the coordinates of (T4), implies $GIP(G)=PSPACE$ for a natural class of theories including theories that are not isomorphic to quantum or classical theory? Is $GIP(\text{boxworld})=PSPACE$? No analogue of the realisation theorem for quantum combs is known for boxworld, so the strategies of a boxworld prover have no known finite description of the kind that underlies the quantum upper bound .

— Beyond IP = PSPACE and QIP = PSPACE: Interactive Proofs in Arbitrary Physical Theories  (2610.00847 - Bharti, 1 Oct 2026) in Section 7, Section “Open questions,” Question “Effectiveness hypotheses” (label q:effective); see also the preceding paragraph in Section 7

What are the multi-prover classes of boxworld and of other theories satisfying (T1)--(T4)?

— Beyond IP = PSPACE and QIP = PSPACE: Interactive Proofs in Arbitrary Physical Theories  (2610.00847 - Bharti, 1 Oct 2026) in Section 7, Section “Open questions,” Question “Several provers” (label q:multi)

Which parts of Theorems~\ref{thm:cl-universal} and~\ref{thm:qm-universal} survive when local tomography is replaced by a weaker structural assumption on composites?

— Beyond IP = PSPACE and QIP = PSPACE: Interactive Proofs in Arbitrary Physical Theories  (2610.00847 - Bharti, 1 Oct 2026) in Section 7, Section “Open questions,” Question “Theories without local tomography” (label q:LT)