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We present a distribution 𝓓 over inputs in ± 1 2 N, 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 Ω (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) /√ 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.
Raz et al. (Wed,) studied this question.