The distribution monad defined on the category of sets can be extended to the category of (locally small) categories, which also turns out to be a monad. This crucial observation gives the notion of a convex category allowing us to place the theory of simplicial distributions in a more abstract categorical setting. In this way, contextuality and related notions, such as contextual fraction, can be studied abstractly. Simplicial distributions have an additional monoid structure absent in the sheaf-theoretic description of non-signaling distributions.Then the contextual fraction of simplicial distributions can be described as a measure of the non-invertibility of monoid elements.

Mermin square scenario is a fundamental and simple example of a contextual linear system. The existence of classical solutions to this linear system depends on the parity of the linear system. In the even case, a classical solution always exists. However, odd parity leads to no classical solution but admits a solution in the 2-qubit Pauli group. Considering all possible non-signaling distributions respecting the parity of the linear system gives a polytope. Up to isomorphism, there are two kinds distinguished by parity. The even parity Mermin polytope only has deterministic vertices, whereas the odd parity Mermin polytope has two kinds of contextual vertices. The former can be utilized to give new proof of the celebrated Fine theorem characterizing noncontextual distributions on the CHSH scenario. The latter is a toy model for a polytope used in classical simulation and can be used to capture the vertices of that more complicated polytope.

An essential distinction between qubits and quantum systems of odd local dimension (qudits) is the structure of the Pauli group. In odd dimensions, this algebraic structure allows for classical solutions to any linear system with a solution in the Pauli group. However, this is not the case for qubits due to the celebrated Mermin square and star linear systems. The state-dependent version of the Mermin star has a further connection to computation. For a contextual state, the resulting computation, given in the measurement-based quantum computation scheme, elevates classical binary computation to classical universal computation. However, this phenomenon fails in odd local dimensions. We provide an example of a group that can give a computational advantage but is noncontextual: Any linear system admitting a solution in this group admits a classical solution.

The theory of simplicial distributions is a framework for studying quantum contextuality that generalizes both the sheaf-theoretic of Brandenburger and Abramsky's approach and the topological approach of Okay et al. In this framework, measurements and outcomes are represented by spaces. The name simplicial comes from the combinatorial objects known as simplicial sets that are used in modern homotopy theory. The notion of contextuality can be generalized to simplicial distributions, and celebrated scenarios such as the CHSH scenario can be realized in various ways. The figure represents the realization of this scenario as a punctured torus. The main benefit of this approach is that the usual ad-hoc arguments relating different sets of inequalities specifying a polytope can be studied using topological intuition.