Interaction in quantum communication and the complexity of set disjointness

One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet in some scenarios there are ways of conveying information with exponentially fewer qubits than possible classically [3, 26]. Moreover, these methods have a very simple structure---they involve only few message exchanges between the communicating parties. We consider the question as to whether every classical protocol may be transformed to a "simpler" quantum protocol---one that has similar efficiency, but uses fewer message exchanges. We show that for any constant k, there is a problem such that its k+1 message classical communication complexity is exponentially smaller than its k message quantum communication complexity, thus answering the above question in the negative. This in particular proves a round hierarchy theorem for quantum communication complexity, and implies via a simple reduction, an Ω(N^(1/k)) lower bound for k message protocols for Set Disjointness for constant k. Our result builds on two primitives, local transitions in bi-partite states (based on previous work) and average encoding which may be of significance in other contexts as well.