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One of the most intriguing fa ts about ommuni ation using quantum states is that these states annot be used to transmit more lassi al bits than the number of qubits used, yet in some s enarios there are ways of onveying information with exponentially fewer qubits than possible lassi ally [3, 26℄. Moreover, these methods have a very simple stru ture|they involve only few message ex hanges between the ommuniating parties. We onsider the question as to whether every lassi al proto ol may be transformed to a \simpler" quantum proto ol|one that has similar eÆ ien y, but uses fewer message ex hanges. We show that for any onstant k, there is a problem su h that its k+1 message lassi al ommuni ation Supported by the EU 5th framework program QAIP IST1999-11234 and by NWO grant 612.055.001. ySupported by Charles Lee Powell Foundation, NSF grant CCR 98761722, and Institute for Quantum Information. This work was done while the author was at University of California, Berkeley, supported by a JSEP grant and NSF grant CCR 9800024, and later at DIMACS Center and AT&T Labs, supported by NSF grants STC 91-19999, CCR 99-06105 and EIA 00-80234. zMost of this work was done while the author was at the University of California at Berkeley, and supported in part by a David and Lu ile Pa kard Fellowship for S ien e and Engineering and NSF NYI Grant CCR-9457799. xThis work was done while the author was on leave at the University of California at Berkeley. Supported in part by a David and Lu ile Pa kard Fellowship for S ien e and Engineering, NSF Grant CCR-9912428, NSF NYI Grant CCR9457799, and an Alfred P. Sloan Resear h Fellowship. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. STOC’01, July 6–8, 2001, Hersonissos, Crete, Greece. Copyright 2001 ACM 1-58113-349-9/01/0007 ...$5.00. omplexity is exponentially smaller than its k message quantum ommuni ation omplexity, thus answering the above question in the negative. This in parti ular proves a round hierar hy theorem for quantum ommuni ation omplexity, and implies via a simple redu tion, an (N1=k) lower bound for k message proto ols for Set Disjointness for onstant k. Our result builds on two primitives, lo al transitions in bipartite states (based on previous work) and average en oding whi h may be of signi an e in other ontexts as well.