The obfuscation of a combinational logic circuit using logic locking is similar to symmetric cryptography and can be expressed in terms of a quantified Boolean function. Satisfiability checking algorithms are used to reveal the secret key of the logic locking (so-called SAT attacks). In this paper, we analyze the impact of quantum computers for these SAT-based attacks similar to the impact on symmetric cryptographic ciphers. For the first time to our knowledge, we propose modifications of existing logic locking schemes to achieve the same security level in a post-quantum era. Furthermore, we investigate dedicated countermeasures against SAT- based algorithms, suggest modifications and evaluate their effectiveness against quantum attacks.