  Mutually unbiased bases (MUBs) are a primitive used in quantum information
processing to capture the principle of complementarity. While constructions of
maximal sets of d+1 such bases are known for systems of prime power dimension
d, it is unknown whether this bound can be achieved for any non-prime power
dimension. In this paper we demonstrate that maximal sets of MUBs come with a
rich combinatorial structure by showing that they actually are the same objects
as the complex projective 2-designs with angle set {0,1/d}. We also give a new
and simple proof that symmetric informationally complete POVMs are complex
projective 2-designs with angle set {1/(d+1)}.
