Some Results on Circuit Lower Bounds and Derandomization of Arthur-Merlin Problems

We prove a downward separation for Σ2-time classes. Specifically, we prove that if Σ2E does not have polynomial size non-deterministic circuits, then Σ2SubEXP does not have fixed polynomial size nondeterministic circuits. To achieve this result, we use Santhanam’s technique [16] on augmented Arthur-Merlin protocols defined by Aydinlioğlu and van Melkebeek [1]. We show that augmented ArthurMerlin protocols with one bit of advice do not have fixed polynomial size non-deterministic circuits. We also prove a weak unconditional derandomization of a certain type of promise Arthur-Merlin protocols. Using Williams’ easy hitting set technique [20], we show that Σ2-promise AM problems can be decided in Σ2SubEXP with n c advice, for some fixed constant c.