The layer complexity of Arthur-Merlin-like communication

In communication complexity the Arthur-Merlin (AM) model is the most natural one that allows both randomness and non-determinism. Presently we do not have any super-logarithmic lower bound for the AM-complexity of an explicit function. Obtaining such a bound is a fundamental challenge to our understanding of communication phenomena. In this article we explore the gap between the known techniques and the complexity class AM. In the first part we define a new natural class, Small-advantage Layered Arthur-Merlin (SLAM), that has the following properties:

• SLAM is (strictly) included in AM and includes all previously known subclasses of AM with non-trivial lower bounds.
• SLAM is qualitatively stronger than the union of those classes.
• SLAM is a subject to the discrepancy bound: in particular, the inner product function does not have an efficient SLAM-protocol. Structurally this can be summarised as SBP $\cup$ UAM $\subset$ SLAM $\subseteq$ AM $\cap$ PP. In the second part we ask why proving a lower bound of $\omega (\sqrt{n})$ on the MA-complexity of an explicit function seems to be difficult. Both of these results are related to the notion of layer complexity, which is, informally, the number of “layers of non-determinism” used by a protocol.