I understand the only multilinear maps used in cryptography are bilinear maps, and higher arity multilinear maps are not "known." Why does the composition of bilinear maps not yield usable higher-arity maps? I thought the primary feature of multilinear maps in cryptography is simply their multilinearity along with non-degeneracy and efficient computability, so despite restricting to a very small class of multilinear maps (those composed of bilinear maps), those three properties are preserved.
Are my assumptions wrong? Would bilinear maps with appropriate signatures for desired compositions be too difficult to find? What am I missing?