Near-Maximum Circuit Lower Bounds for Exponential Time with Merlin-Arthur Queries

Hanlin Ren; Ryan Williams

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arXiv · 2607.09963

Near-Maximum Circuit Lower Bounds for Exponential Time with Merlin-Arthur Queries

Hanlin Ren·Ryan Williams

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Abstract

We prove a near-maximum ($2^n / n$) circuit lower bound for the complexity class $\mathsf{E}^{\mathrm{pr}\mathsf{MA}}/_1$, corresponding to exponential time with access to a promise-$\mathsf{MA}$ oracle and one bit of advice. Our proof incorporates the iterative win-win paradigm (Chen--Lu--Oliveira--Ren--Santhanam, FOCS'23), the reduction from the Range Avoidance problem to circuit lower bounds (Je\v{r}\'abek, Ann. Pure Appl. Log. '04; Korten, FOCS'21), and the PCP theorem. Crucial to our proof is the analysis of the complexity class $\mathsf{P}^\mathsf{NP}[{\textsf{#rounds}}=r, {\textsf{length}}=s]$, which is $\mathsf{P}^\mathsf{NP}$ with $r(n)$ adaptive rounds of $\mathsf{NP}$ queries, where each $\mathsf{NP}$ query has witness length $s(n)$.

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Hanlin Ren, Ryan Williams. 2026-07-10. Near-Maximum Circuit Lower Bounds for Exponential Time with Merlin-Arthur Queries. https://arxiv.org/abs/2607.09963

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