This is joint work with Paul Beame, Matei David, and Toni Pitassi.

We solve some fundamental problems in the number-on-forehead (NOF) $k$-player communication model. We show that there exists a function which has at most logarithmic communication complexity for randomized protocols with a one-sided error probability of 1/3 but which has linear communication complexity for deterministic protocols. The result is true for $k=n^{O(1)}$ players, where $n$ is the number of bits on each players' forehead. This separates the analogues of RP and P in the NOF communication model.

We also show that there exists a function which has constant randomized complexity for public coin protocols but at least logarithmic complexity for private coin protocols. No larger gap between private and public coin protocols is possible. Our lower bounds are existential and we do not know of any explicit function which allows such separations. However, for the 3-player case we exhibit an explicit function which has $\Omega(\log\log n)$ randomized complexity for private coins but only constant complexity for public coins.