Simplicial techniques in quantum foundations

Probability distributions produced by quantum measurements are more general than the ones produced by classical measurements. This is due to the fact that only commuting observables, for example represented by a set of commuting unitaries, can be simultaneously measured. Nonclassical properties of these distributions can be studied using classifying spaces assembled from commuting unitary operators. In this talk I will describe recent applications of these classifying spaces to the study of certain problems in quantum foundations which have also important applications in quantum computing.