Security mechanisms of Electronic Personal Documents (eCards) depend on (asymmetric) cryptography that is and always has been subject to the threat of compromise, be it from conventional attacks or quantum computers. With Post-Quantum Cryptography (PQC), we now have alternative building blocks at hand that can be leveraged to protect against both kind of attacks. Thus, PQC should be incorporated into eCard ecosystems, yet it is not clear how this is done best. In the work at hand, we review the state of currently used crypto-systems for eCard security, as well as their possible quantum-secure replacements. Further, we identify and categorize respective challenges that need to be addressed, present and assess existing approaches for their solution, and formulate research questions for open issues. By providing an overview of the situation, we help unraveling the issue and pave the way toward quantum-safe electronic Identity Documents and electronic Machine-Readable Travel Documents.