A quantum computer running Shor's algorithm would be famously useful for decrypting information encrypted by many classical public-key cryptography algorithms. Is there any reason (either a specific proposed protocol or a general heuristic argument) to suspect that a quantum computer could be useful for encrypting information in a way that's more secure than is possible with a classical computer with comparable resources? Ideally, secure even against attacks by quantum computers?
I'm not talking about "physics-based" quantum encryption schemes like quantum key distribution (discussed here) or other schemes that require transmitting a coherent quantum state over a quantum channel. I'm talking about more traditional "mathematics-based" encryption schemes, in which one or both parties have a quantum computer at each end, but they can only transmit the encoded information in the form of classical bit strings over an insecure classical channel (potentially after having transmitted a symmetric key out of band).
This question is inspired by Scott Aaronson's comments here and here on his blog. Apparently people regularly claim that quantum computers (not QKD) could be useful for encryption, but Prof. Aaronson has never understood why.