Quantum Polymorphisms and Commutativity Gadgets

We introduce the concept of quantum polymorphisms to the complexity theory of non-local games. We use this notion to give a full characterisation of the existence of commutativity gadgets for relational structures, introduced by Ji as a method for achieving quantum soundness of classical CSP reductions. Prior to our work, a classification was only known in the Boolean case [Culf--Mastel, STOC'25]. As an application of our framework, we prove that the entangled CSP parameterised by odd cycles is undecidable. Furthermore, we establish a quantum version of Galois connection for entangled CSPs in the case of non-oracular quantum homomorphisms.