Abstract: With the potential advent of quantum computers, cryptographers are searching for problems that are hard even if an adversary possesses a quantum computer. We'll analyze one such candidate, the supersingular isogeny problem on elliptic curves: given two (supersingular) elliptic curves, compute an isogeny between them. We'll review the literature for evidence that this problem is hard for quantum adversaries, and give an application to Diffie-Hellman key exchange.