The geometry of simplicial distributions on suspension scenarios

Quantum measurements often exhibit non-classical features, such as contextuality. Recently, simplicial distributions have been introduced as a framework that captures these phenomena. In this work, we study the geometrical structure of simplicial distributions using topological methods, leveraging the fact that, in this simplicial framework, measurements and outcomes are represented as spaces. Applying the cone construction to the measurement space shows that the corresponding non-signaling polytope equals the join of m copies of the original polytope, where m is the number of possible outcomes per measurement. Gluing two copies of cone measurement spaces yields a suspension measurement space, and the decompositions performed for simplicial distributions on these spaces provide insights into contextuality on suspensions. We further use these results to derive a new class of Bell inequalities and contextual vertices, providing a mathematical explanation for certain contextual vertices previously reported in the literature.