In recent work we are revisiting the oft-neglected recursive Fourier sampling (RFS) problem, introduced by Bernstein and Vazirani  to prove an oracle separation between BPP and BQP. RFS seems to offer the best hope for proving that BQP is not in the polynomial hierarchy relative to an oracle. As a first step, we seek to place RFS outside a smaller complexity class such as AM or Σ2P relative to an oracle.
In previous work , one of us showed that the known quantum algorithm for RFS is essentially optimal, despite its seemingly wasteful need to uncompute information. This implies that, to place BQP outside of PH[log] relative to an oracle, one would need to go outside the RFS framework. Our proof argues that, given any variant of RFS, either the adversary method of Ambainis  yields a good quantum lower bound, or else there is an efficient classical algorithm. This technique may be of independent interest.