A core goal of the NIST PQC competition is to produce public-key encryption (PKE) schemes which, even if attacked with a large-scale quantum computer, maintain the security guarantees needed by applications. The main security focus in the NIST PQC context has been IND-CCA security,but other applications demand that PKE schemes provide anonymity (Bellare et al., ASIACRYPT 2001), and robustness (Abdalla et al., TCC 2010). Examples of such applications include anonymous communications systems, cryptocurrencies, anonymous credential systems, searchable encryption, and auction protocols. Almost nothing is known about how to build post-quantum PKE schemes offering these security properties. In particular, the status of the NIST PQC finalists with respect to anonymity and robustness is unknown.