This thesis presents two new approaches to two topics in quantum foundations: Part I: The study of causal structures and higher order processes, i.e., transformations of processes.Part II: The study of non-locality and contextuality using effect algebras.embed in a category of presheaves and this allows us to make a connection between the two approaches using adjunctions.In particular, we are able to relate our effect algebraic framework to the established presheaf approach of Abramsky and Brandenburger.In a similar fashion we relate effect algebras to test spaces.We describe explicitly the examples of Bell, Kochen-Specker, Hardy and GHZ.