  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.
