The quantum-to-classical graph homomorphism game

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