Is Diffie-Hellman post-quantum secure? If Alice and Bob have a secure channel for key exchange and Mallory doesn't man-in-the-middle attack them, but in the future eavesdrops on their connection and sees the key exchange, can Mallory break it like RSA with a quantum computer?
A quantum computer solves the discrete logarithm problem for both finite fields and elliptic curves. Being able to efficiently calculate discrete logarithms implies being able to break Diffie-Hellman, so Diffie-Hellman on either of them is not secure against an adversary who owns a large quantum computer.

There might be other groups in which DL problem is hard even against a quantum computer, but I've never heard of such a group.