The chromatic number of a graph is the least number of colors that one needs to color vertices in such a way that adjacent vertices are always different colors. The graph coloring game is a game played by two persons and a referee. If the players are isolated and only allowed a traditional binary computer, then the fewest number of colors needed to convince the referee that they have an actual coloring of the graph is exactly the chromatic number. However, if they are both allowed to use quantum computers, then they can convince a referee with probability one that they have a coloring using far fewer colors. This number is known as the quantum chromatic number. In this talk we will outline what is known about the quantum chromatic number and its behavior. In joint work with I. Todorov, we have shown that the value of the quantum chromatic number could vary depending on the axioms that you use for quantum mechanics. We study the properties of these potentially different quantum chromatic numbers. For example, we prove that if one uses relativistic quantum mechanics then the quantum chromatic number is solvable by an SDP. Whether or not the non-relativistic quantum chromatic number is solvable by an SDP in still open, but it is known to be an NP-hard problem.