Quasi-linear 2-query PCPPs with small soundness or ruling them out

Determine whether 2-query probabilistically checkable proofs of proximity (PCPPs) with quasi-linear size, constant (non-Boolean) alphabet size, and small soundness error exist for general circuit-satisfaction languages SAT(φ). Either construct such 2-query PCPPs or establish lower bounds that rule out their existence under these parameters.

Background

The paper constructs 3-query PCPPs with quasi-linear size and small soundness by leveraging new decodable PCPs and composition techniques. However, known lower bounds do not preclude the possibility of achieving similar soundness with only 2 queries.

The authors explicitly leave open whether such stronger 2-query PCPPs exist, or whether new lower bounds can exclude them, highlighting a central gap between current constructions (3 queries) and the best conceivable query complexity (2 queries).

References

Existing lower bounds for PCPPs do not rule out quasi-linear $2$-query PCPPs with constant (but not Boolean) alphabet size that have small soundness, and we leave the problems of giving better $2$-query PCPP constructions or lower bounds to future research.

— 3-Query RLDCs are Strictly Stronger than 3-Query LDCs (2512.12960 - Gur et al., 15 Dec 2025) in Discussion and Open Problems