What does "post-quantum" actually mean for a Bitcoin L2 when settlement still happens on a non-PQ L1?

The discussion revolves around the security challenges and considerations necessary when integrating post-quantum resistant layer 2 (L2) solutions onto a quantum-vulnerable layer 1 (L1) blockchain. It's essential to address whether these L2 solutions can effectively authenticate user deposits and attribute them correctly, even if the initial peg-in is secured against quantum threats. The authentication process and the proof systems used are critical, as vulnerabilities could allow attackers to forge proofs and authorize illegitimate transactions, potentially draining funds from the bridge.

One proposed solution involves the use of one-time signature (OTS) schemes, which could offer enhanced security. However, merely incorporating an OTS into existing structures like Bitcoin Script is insufficient. It's crucial that the signature verification processes are meticulously designed to handle the nuances of post-quantum security.

There's also a concern regarding the practical implementation of such technologies in scenarios where the elliptic curve digital signature algorithm (ECDSA) is compromised. Despite the potential for L2 solutions to validate transactions using post-quantum secure primitives, challenges remain if the underlying assets are still controlled by EC keys. This control issue raises doubts about the effectiveness of declaring transactions secure or authorized under quantum-threatened mechanisms, highlighting the complexity of securely integrating post-quantum cryptography into existing blockchain frameworks.