Abstract: We introduce a game in which two cooperating parties attempt to convince a referee that two graphs G and H are isomorphic. Classical strategies that win this game correspond to actual isomorphisms of G and H. However, if the two parties are given access to certain quantum mechanical resources (local measurements on a shared entangled state), then they can sometimes win this game even when G and H are not isomorphic. This operationally defined notion of quantum isomorphism turns out to have an elegant algebraic description in terms of magic unitaries, a notion from the theory of quantum groups. Moreover, quantum isomorphism can be completely reformulated in terms of the quantum automorphism group of a graph. We will discuss how these connections allow us to prove that graphs G and H are quantum isomorphic if and only if they admit the same number of homomorphisms from any *planar* graph. This can be viewed as a quantum analog of a classical result of Lovasz from over 50 years ago: graphs G and H are isomorphic if and only if they admit the same number homomorphisms from any graph. Though the connection to quantum groups is crucial, the details of the proof of this result are mostly combinatorial. Please email Jane Bergman (firstname.lastname@example.org) or Jozsef Balogh (email@example.com) for Zoom meeting link.