The Two Queries Assumption and Arthur-Merlin Classes

We explore the implications of the two queries assumption, P [1] = P SAT [2] || , with respect to the polynomial hierarchy (PH) and Arthur-Merlin classes. We prove the following results under the assumption P [1] = P SAT [2] || : 1. AM = MA 2. There exists no relativizable proof for PH ⊆ MA 3. Every problem in PH can be solved by a non-uniform variant of a Merlin-Arthur(MA) protocol where Arthur(the verifier) has access to one bit of advice. 4. PH = P SAT [1],MA[1] || Under the two queries assumption, Chakaravarthy and Roy showed that PH collapses to NO2 [5]. Since NP ⊆ MA ⊆ NO2 unconditionally, our result on relativizability improves upon the result by Buhrman and Fortnow that we cannot show that PH ⊆ NP using relativizable proof techniques [3]. However, we show a containment of PH in a non-uniform version of MA where Arthur has one bit of advice. This also improves upon the result by Kadin that PH ⊂ NP/poly [11]. Our fourth result shows that simulating MA in a P [1] machine is as hard as collapsing PH to P .