Two queries

We consider the question whether two queries to SAT are as powerful as one query. We show that if P^NP[1]=P^NP[2] then; locally either NP=coNP or NP has polynomial-size circuits; P^NP=P^NP[1]; Σ₂^p=UP^NP[1]∩RP^NP[1] ; PH=BPP^NP[1]. Moreover we extend work of E. Hemaspaandra et al. (1997) to show that if P(Σ₂^p [1])=P(Σ₂^p[2]) then Σ₂^p=Π₂^p. We also give a relativized world where P^NP[1]=P^NP[2] but NP≠coNP