  We show several results related to interactive proof modes of communication
complexity. First we show lower bounds for the QMA-communication complexity of
the functions Inner Product and Disjointness. We describe a general method to
prove lower bounds for QMA-communication complexity, and show how one can
'transfer' hardness under an analogous measure in the query complexity model to
the communication model using Sherstov's pattern matrix method. Combining a
result by Vereshchagin and the pattern matrix method we find a communication
problem with AM-communication complexity $O(\log n)$, PP-communication
complexity $\Omega(n^{1/3})$, and QMA-communication complexity
$\Omega(n^{1/6})$. Hence in the world of communication complexity
noninteractive quantum proof systems are not able to efficiently simulate
co-nondeterminism or interaction. These results imply that the related
questions in Turing machine complexity theory cannot be resolved by
'algebrizing' techniques. Finally we show that in MA-protocols there is an
exponential gap between one-way protocols and two-way protocols (this refers to
the interaction between Alice and Bob). This is in contrast to
nondeterministic, AM-, and QMA-protocols, where one-way communication is
essentially optimal.
