More details on the content of this entry are at Bohr topos.
There are several approaches using topos theory to formulate and study foundations of quantum mechanics. Some people say “quantum topos theory”, which is however a bit unfortunate as it is not about some quantum analogue of a topos (while such an object may be desired, say in noncommutative geometry), but about using the usual topoi to model quantum mechanics.
The main proposal is from Chris Isham and his school, see higher category theory and physics.
Andread Doering, C.J. Isham, A Topos Foundation for Theories of Physics I-IV quant-ph/0703060
Kunji Nakayama, Sheaves in quantum topos induced by quantization, arxiv/1109.1192
See also Kochen-Specker theorem for the topos theoretic interpretation summarizing
Another, more recent variant of using topos theory in quantum foundations is due Nijmengen school, see Bohr topos for detail.