  We investigate the action of the first three levels of the Clifford hierarchy
on sets of mutually unbiased bases comprising the Ivanovic MUB and the Alltop
MUBs. Vectors in the Alltop MUBs exhibit additional symmetries when the
dimension is a prime number equal to 1 modulo 3 and thus the set of all Alltop
vectors splits into three Clifford orbits. These vectors form configurations
with so-called Zauner subspaces, eigenspaces of order 3 elements of the
Clifford group highly relevant to the SIC problem. We identify Alltop vectors
as the magic states that appear in the context of fault-tolerant universal
quantum computing, wherein the appearance of distinct Clifford orbits implies a
surprising inequivalence between some magic states.
