  Given a prior probability distribution over a set of possible oracle
functions, we define a number of queries to be useless for determining some
property of the function if the probability that the function has the property
is unchanged after the oracle responds to the queries. A familiar example is
the parity of a uniformly random Boolean-valued function over $\{1,2,...,N\}$,
for which $N-1$ classical queries are useless. We prove that if $2k$ classical
queries are useless for some oracle problem, then $k$ quantum queries are also
useless. For such problems, which include classical threshold secret sharing
schemes, our result also gives a new way to obtain a lower bound on the quantum
query complexity, even in cases where neither the function nor the property to
be determined is Boolean.
