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The first separation between quantum polynomial time and classicalbounded-error polynomial time was due to Bernstein and Vazirani in 1993. Theyfirst showed a O(1) vs. Omega(n) quantum-classical oracle separation based onthe quantum Hadamard transform, and then showed how to amplify this into an^{O(1)} time quantum algorithm and a n^{Omega(log n)} classical query lowerbound.  We generalize both aspects of this speedup. We show that a wide class ofunitary circuits (which we call dispersing circuits) can be used in place ofHadamards to obtain a O(1) vs. Omega(n) separation. The class of dispersingcircuits includes all quantum Fourier transforms (including over nonabeliangroups) as well as nearly all sufficiently long random circuits. Second, wegive a general method for amplifying quantum-classical separations that allowsus to achieve a n^{O(1)} vs. n^{Omega(log n)} separation from any dispersingcircuit.

Superpolynomial speedups based on almost any quantum circuit