  In this thesis we study symmetric structures in Hilbert spaces known as
symmetric informationally complete positive operator-valued measures
(SIC-POVMs), mutually unbiased bases (MUBs), and MUB-balanced states. Our tools
include symmetries such as the Weyl-Heisenberg (WH) group symmetry, Clifford
unitaries, Zauner symmetry, and Galois-unitaries (g-unitaries), which are
non-linear operators defined to generalize the notion of anti-unitaries.
