Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates, and thus a potential way to use these quantum devices is using a hybrid scheme that interleaves them with classical computers. For example, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Along the line, it seems possible that a general quantum computer may only be polynomially faster than a hybrid quantum-classical computer. Jozsa raised the question of whether (BQP = BPP^{BQNC}) and conjectured that they are equal, where (BQNC) means (polylog)-depth quantum circuits. Nevertheless, Aaronson conjectured an oracle separation for these two classes and gave a candidate. In this work, we prove Aaronson’s conjecture for a different but related oracle problem. Our result also proves that Jozsa’s conjecture fails relative to an oracle.