  We compare classical and quantum query complexities of total Boolean
functions. It is known that for worst-case complexity, the gap between quantum
and classical can be at most polynomial. We show that for average-case
complexity under the uniform distribution, quantum algorithms can be
exponentially faster than classical algorithms. Under non-uniform distributions
the gap can even be super-exponential. We also prove some general bounds for
average-case complexity and show that the average-case quantum complexity of
MAJORITY under the uniform distribution is nearly quadratically better than the
classical complexity.
