Connecting XOR and XOR* games

In this work we focus on two classes of games: XOR nonlocal games and XOR* sequential games with monopartite resources. XOR games have been widely studied in the literature of nonlocal games, and we introduce XOR* games as their natural counterpart within the class of games where a resource system is subjected to a sequence of controlled operations and a final measurement. Examples of XOR* games are $2\rightarrow 1$ quantum random access codes (QRAC) and the CHSH* game introduced by Henaut et al. in [PRA 98,060302(2018)]. We prove, using the diagrammatic language of process theories, that under certain assumptions these two classes of games can be related via an explicit theorem that connects their optimal strategies, and so their classical (Bell) and quantum (Tsirelson) bounds. We also show that two of such assumptions -- the reversibility of transformations and the bi-dimensionality of the resource system in the XOR* games -- are strictly necessary for the theorem to hold by providing explicit counterexamples. We conclude with several examples of pairs of XOR/XOR* games and by discussing in detail the possible resources that power the quantum computational advantages in XOR* games.
Comment: 15 pages double column, 2 figures/diagrams. Typos corrected, conclusions updated