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We study the bipartite composition of elementary toy systems with statespaces described by regular polygons. We provide a systematic method tocharacterize the entangled states in the maximal tensor product composition ofsuch systems. Applying this method, we show that while a bipartite pentagonsystem allows two and exactly two different classes of entangled states, in thehexagon case, there are exactly six different classes of entangled states. Wethen prove a generic no-go result that the maximally entangled state for anybipartite odd gon system does not depict Hardy's nonlocality behaviour.However, such a state for even gons exhibits Hardy's nonlocality, and in thatcase, the optimal success probability decreases with the increasing number ofextreme states in the elementary systems. Optimal Hardy's success probabilityfor the non-maximally entangled states is also studied that establishes thepresence of beyond quantum correlation in those systems, although the resultingcorrelation lies in the almost quantum set. Furthermore, it has been shown thatmixed states of these systems, unlike the two-qubit case, can depict Hardy'snonlocality behaviour which arises due to a particular topological feature ofthese systems not present in the two-qubit system.

Bipartite polygon models: entanglement classes and their nonlocal behaviour