  We consider the following problem: Given a set S of at most n elements from a
universe of size m, represent it in memory as a bit string so that membership
queries of the form "Is x in S?" can be answered by making at most t probes
into the bit string. Let s(m,n,t) be the minimum number of bits needed by any
such scheme. We obtain new upper bounds for s(m,n,t=2), which match or improve
all the previously known bounds. We also consider the quantum version of this
problem and obtain improved upper bounds.
