Entangled Simultaneity Versus Classical Interactivity in Communication Complexity

In 1999 Raz demonstrated a partial function that had an efficient quantum two-way communication protocol but no efficient classical two-way protocol and asked whether there existed a function with an efficient quantum one-way protocol, but still no efficient classical two-way protocol. In 2010 Klartag and Regev demonstrated such a function and asked whether there existed a function with an efficient quantum simultaneous-messages protocol, but still no efficient classical two-way protocol. In this work we answer the latter question affirmatively and present a partial function Shape that can be computed by a protocol sending entangled simultaneous messages of poly-logarithmic size, and whose classical two-way complexity is lower bounded by a polynomial.