Is it possible to MAC a quantum state with a classical key under reasonable assumption? Assume that classical one-way functions secure against quantum adversaries exist.  Is it possible, given a quantum state $Q$ and classical secret key $k$, produce a quantum state $AuthQ$ such that:


One who knows $k$ can recover $Q$, provided that $AuthQ$ has not been altered.
If $AuthQ$ has been altered, this can be detected with high probability.


It may or may not be possible to recover the original state without the secret key.  What matters is that, given the secret key, one can be confident that:


The creator of the quantum state had the secret key
The message has not been tampered with since.
Howard Barnum, Claude Crepeau, Daniel Gottesman, Adam Smith, Alain Tapp. "Authentication of Quantum Messages", FOCS 2002.

http://www.cse.psu.edu/~ads22/pubs/PS-CSAIL/BCGST02-focs-final.pdf

As with encryption, there is a protocol that requires no computational assumptions. It uses a key of length about 2n+2k to authenticate n qubits with security level 2^{-k}.