Exponential separation of quantum and classical non-interactive multi-party communication complexity

We give the first exponential separation between quantum and classical multi-party communication complexity in the (non-interactive) one-way and simultaneous message passing settings. For every k, we demonstrate a relational communication problem between k parties that can be solved exactly by a quantum simultaneous message passing protocol of cost O (log n) and requires protocols of cost n(c/k2), where c > 0 is a constant, in the classical non-interactive one-way message passing model with shared randomness and bounded error Thus our separation of corresponding communication classes is superpolynomial as long as k = o(root log n/log log n) and exponential for k = O (1).