  We address the question whether quantum memory is more powerful than
classical memory. In particular, we consider a setting where information about
a random n-bit string X is stored in r classical or quantum bits, for r<n,
i.e., the stored information is bound to be only partial. Later, a randomly
chosen binary question F about X is asked, which has to be answered using only
the stored information. The maximal probability of correctly guessing the
answer F(X) is then compared for the cases where the storage device is
classical or quantum mechanical, respectively.
  We show that, despite the fact that the measurement of quantum bits can
depend arbitrarily on the question F to be answered, the quantum advantage is
negligible already for small values of the difference n-r.
  An implication for cryptography is that privacy amplification by application
of a compression function mapping n-bit strings to s-bit strings (for some
s<n-r), chosen publicly from a two-universal class of hash functions, remains
essentially equally secure when the adversary's memory is allowed to be r
quantum rather than only r classical bits.

Quantum Physics