It is a longstanding open problem to devise an oracle relative to which BQP

does not lie in the Polynomial-Time Hierarchy (PH). We advance a natural

conjecture about the capacity of the Nisan-Wigderson pseudorandom generator

[NW94] to fool AC_0, with MAJORITY as its hard function. Our conjecture is

essentially that the loss due to the hybrid argument (which is a component of

the standard proof from [NW94]) can be avoided in this setting. This is a

question that has been asked previously in the pseudorandomness literature

[BSW03]. We then make three main contributions: (1) We show that our conjecture

implies the existence of an oracle relative to which BQP is not in the PH. This

entails giving an explicit construction of unitary matrices, realizable by

small quantum circuits, whose row-supports are "nearly-disjoint." (2) We give a

simple framework (generalizing the setting of Aaronson [A10]) in which any

efficiently quantumly computable unitary gives rise to a distribution that can

be distinguished from the uniform distribution by an efficient quantum

algorithm. When applied to the unitaries we construct, this framework yields a

problem that can be solved quantumly, and which forms the basis for the desired

oracle. (3) We prove that Aaronson's "GLN conjecture" [A10] implies our

conjecture; our conjecture is thus formally easier to prove. The GLN conjecture

was recently proved false for depth greater than 2 [A10a], but it remains open

for depth 2. If true, the depth-2 version of either conjecture would imply an

oracle relative to which BQP is not in AM, which is itself an outstanding open

problem. Taken together, our results have the following interesting

interpretation: they give an instantiation of the Nisan-Wigderson generator

that can be broken by quantum computers, but not by the relevant modes of

classical computation, if our conjecture is true.