Oracle Separation of BQP and PH

We present a distribution D over inputs in {+/- 1}(2N), such that: (1) There exists a quantum algorithm that makes one (quantum) query to the input, and runs in time O(log N), that distinguishes between D and the uniform distribution with advantage Omega(1/log N). (2) No Boolean circuit of quasi-polynomial size and constant depth distinguishes between D and the uniform distribution with advantage better than polylog(N)/root N. By well-known reductions, this gives a separation of the classes Promise-BQP and Promise-PH in the black-box model and implies an oracle relative to which BQP is not contained in PH.