We study shared randomness in the context of multi-party number-in-hand communication protocols in the simultaneous message passing model. We show that with three or more players, shared randomness exhibits new interesting properties that have no direct analogues in the two-party case.

First, we demonstrate a hierarchy of modes of shared randomness, with the usual shared randomness where all parties access the same random string as the strongest form in the hierarchy. We show exponential separations between its levels, and some of our bounds may be of independent interest. For example, we show that the equality function can be solved by a protocol of constant length using the weakest form of shared randomness, which we call "XOR-shared randomness."

Second, we show that quantum communication cannot replace shared randomness in the k-party case, where k >= 3 is any constant. We demonstrate a promise function GP_k that can be computed by a classical protocol of constant length when (the strongest form of) shared randomness is available, but any quantum protocol without shared randomness must send $n^{\Omega(1)}$ qubits to compute it. Moreover, the quantum complexity of GP_k remains $n^{\Omega(1)}$ even if the "second strongest" mode of shared randomness is available. While a somewhat similar separation was already known in the two-party case, in the multi-party case our statement is qualitatively stronger:

* In the two-party case, only a relational communication problem with similar properties is known.

* In the two-party case, the gap between the two complexities of a problem can be at most exponential, as it is known that $2^{O(c)} \log n$ qubits can always replace shared randomness in any c-bit protocol. Our bounds imply that with quantum communication alone, in general, it is not possible to simulate efficiently even a three-bit three-party classical protocol that uses shared randomness.