Optimal quantum query bounds for almost all Boolean functions
  We show that almost all n-bit Boolean functions have bounded-error quantum query complexity at least n/2, up to lower-order terms. This improves over an earlier n/4 lower bound of Ambainis, and shows that van Dam's oracle interrogation is essentially optimal for almost all functions. Our proof uses the fact that the acceptance probability of a T-query algorithm can be written as the sum of squares of degree-T polynomials. 
Andris Ambainis (U of Latvia), Arturs Backurs (U of Latvia), Juris   Smotrovs (U of Latvia), Ronald de Wolf (CWI and U of Am...
quant-ph