In this paper we extend a key result of Nisan and Wigderson (J. Comput. System Sci. 49 (1994) 149–167) to the nondeterministic setting: for all α>0 we show that if there is a language in E = DTIME (2 ^{O (n)}) that is hard to approximate by nondeterministic circuits of size 2 ^αn, then there is a pseudorandom generator that can be used to derandomize BP · NP (in symbols, BP · NP = NP ). By applying this extension we are able to answer some open questions in Lutz (Theory Comput. Systems 30 (1997) 429–442) regarding the derandomization of the classes BP · Σ ^P _k and BP · Θ ^P _k under plausible measure theoretic assumptions. As a consequence, if Θ ^P ₂ does not have p-measure 0 , then AM ∩ coAM is low for Θ ^P ₂. Thus, in this case, the graph isomorphism problem is low for Θ ^P ₂. By using the Nisan–Wigderson design of a pseudorandom generator we unconditionally show the inclusion MA ⊆ ZPP ^NP and that MA ∩ coMA is low for ZPP ^NP .