We briefly introduce quantum XOR games with two parties. The main example we will consider is closely related to the notion of embezzling entanglement; i.e., trying to find unitaries $U,V \in M_2(\mathcal{B}(\mathcal{H}))$ in some commuting operator framework, and a state $\psi \in \mathcal{H}$ such that $$(U \otimes V)(e_0 \otimes \psi \otimes e_0)=\frac{1}{\sqrt{2}} (e_0 \otimes \psi \otimes e_0+e_1 \otimes \psi \otimes e_1).$$
Looking at the strategies for quantum XOR games will motivate a study of unitary correlation sets, which arise from states on certain tensor products of a fixed universal operator system. We will see how these sets and the class of quantum XOR games relate to Connes' embedding problem; moreover, embezzlement of entanglement allows us to show that the spatial unitary correlations never form a closed set.