We study a class of quantum measurements that furnish probabilistic
representations of finite-dimensional quantum theory. The Gram matrices
associated with these Minimal Informationally Complete quantum measurements
(MICs) exhibit a rich structure. They are "positive" matrices in three
different senses, and conditions expressed in terms of them have shown that the
Symmetric Informationally Complete measurements (SICs) are in some ways optimal
among MICs. Here, we explore MICs more widely than before, comparing and
contrasting SICs with other classes of MICs, and using Gram matrices to begin
the process of mapping the territory of all MICs. Moreover, the Gram matrices
of MICs turn out to be key tools for relating the probabilistic representations
of quantum theory furnished by MICs to quasi-probabilistic representations,
like Wigner functions, which have proven relevant for quantum computation.
Finally, we pose a number of conjectures, leaving them open for future work.