It has been thought for many years now that some Quantum Information Theoretical problems seem to have strong connections to certain problems about combinatorial geometries. For instance, it has been conjectured that Zauner's conjecture (on the dimension of complex Hilbert spaces which admit maximal sets of mutually unbiased bases (MUBs)) is closely related to the famous prime power conjecture for finite projective planes. (See for instance "Quantum measurements and finite geometry" by William Wootters (Found. Phys., 2006).)

In this talk, which I want to aim at a broad audience, I plan to describe some instances of this theory, including:

I. MUBs,

II. Quantum Theories defined over other rings than the field of complex numbers, and

III. applications of the latter theory on the level of no-cloning results and quantum coding.