  We consider the class of functions whose value depends only on the
intersection of the input X_1,X_2, ..., X_t; that is, for each F in this class
there is an f_F: 2^{[n]} \to {0,1}, such that F(X_1,X_2, ..., X_t) = f_F(X_1
\cap X_2 \cap ... \cap X_t). We show that the t-party k-round communication
complexity of F is Omega(s_m(f_F)/(k^2)), where s_m(f_F) stands for the
`monotone sensitivity of f_F' and is defined by s_m(f_F) \defeq max_{S\subseteq
[n]} |{i: f_F(S \cup {i}) \neq f_F(S)|. For two-party quantum communication
protocols for the set disjointness problem, this implies that the two parties
must exchange Omega(n/k^2) qubits. For k=1, our lower bound matches the
Omega(n) lower bound observed by Buhrman and de Wolf (based on a result of
Nayak, and for 2 <= k <= n^{1/4}, improves the lower bound of Omega(sqrt{n})
shown by Razborov. (For protocols with no restrictions on the number of rounds,
we can conclude that the two parties must exchange Omega(n^{1/3}) qubits. This,
however, falls short of the optimal Omega(sqrt{n}) lower bound shown by
Razborov.)
