Quantum versus classical simultaneity in communication complexity

We present a bipartite partial function c̃Eq-negT , whose communication complexity is O ( (logn) ) in the model of quantum simultaneous message passing (Q‖,pub) and Ω̃(√n) in the model of randomised simultaneous message passing (R‖,pub). In fact, our function has a poly-logarithmic protocol even in the (restricted) model of quantum simultaneous message passing without shared randomness (Q‖), thus witnessing the possibility of qualitative advantage of Q‖ over R‖,pub . This can be interpreted as the strongest known – as of today – example of “super-classical” capabilities of the weakest studied model of quantum communication. The closest previously-known result was given by Buhrman, Cleve, Watrous and de Wolf in 2001: they have shown that the equality function had a protocol of logarithmic cost in Q‖, while its complexity in the model of classical simultaneous message passing without shared randomness (R‖) had already been known to be in Ω(√n). R‖ and Q‖ are “purposely weakened” implementations of simultaneous message passing in communication complexity, which are not closed with respect to mixed strategies (R‖,pub and Q‖,pub can be viewed as the respective “closures”).