  We prove a near optimal round-communication tradeoff for the two-party
quantum communication complexity of disjointness. For protocols with $r$
rounds, we prove a lower bound of $\tilde{\Omega}(n/r + r)$ on the
communication required for computing disjointness of input size $n$, which is
optimal up to logarithmic factors. The previous best lower bound was
$\Omega(n/r^2 + r)$ due to Jain, Radhakrishnan and Sen [JRS03]. Along the way,
we develop several tools for quantum information complexity, one of which is a
lower bound for quantum information complexity in terms of the generalized
discrepancy method. As a corollary, we get that the quantum communication
complexity of any boolean function $f$ is at most $2^{O(QIC(f))}$, where
$QIC(f)$ is the prior-free quantum information complexity of $f$ (with error
$1/3$).
