Communication complexity of entanglement assisted multi-party computation

We consider a quantum and classical version multi-party function computation problem with $n$ players, where players $2, \dots, n$ need to communicate appropriate information to player 1, so that a "generalized" inner product function with an appropriate promise can be calculated. The communication complexity of a protocol is the total number of bits that need to be communicated. When $n$ is prime and for our chosen function, we exhibit a quantum protocol (with complexity $(n-1) \log n$ bits) and a classical protocol (with complexity $(n-1)^2 (\log n^2$) bits). In the quantum protocol, the players have access to entangled qudits but the communication is still classical. Furthermore, we present an integer linear programming formulation for determining a lower bound on the classical communication complexity. This demonstrates that our quantum protocol is strictly better than classical protocols.