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Abstract We compare the algebras of the quantum automorphism group of finite-dimensional C$$^\ast $$-algebra $$B$$, which includes the quantum permutation group $$S_N^+$$, where $$N = \dim B$$. We show that matrix amplification and crossed products by trace-preserving actions by a finite Abelian group $$\Gamma $$ lead to isomorphic $$\ast $$-algebras. This allows us to transfer various properties such as inner unitarity, Connes embeddability, and strong $$1$$-boundedness between the various algebras associated with these quantum groups. 

Motivated by non-local games and quantum coloring problems, we introduce a graph homomorphism game between quantum graphs and classical graphs. This game is naturally cast as a “quantum–classical game,” that is, a non-local game of two players involving quantum questions and classical answers. This game generalizes the graph homomorphism game between classical graphs. We show that winning strategies in the various quantum models for the game is an analog of the notion of non-commutative graph homomorphisms due to Stahlke [IEEE Trans. Inf. Theory 62(1), 554–577 (2016)]. Moreover, we present a game algebra in this context that generalizes the game algebra for graph homomorphisms given by Helton et al. [New York J. Math. 25, 328–361 (2019)]. We also demonstrate explicit quantum colorings of all quantum complete graphs, yielding the surprising fact that the algebra of the four coloring game for a quantum graph is always non-trivial, extending a result of Helton et al. [New York J. Math. 25, 328–361 (2019)].