It seems you can get TLS 1.3 (or atlest slighty modified 1.3) to be quantum secure, but it increases the handshake size by roughly 9x. Cloudflare unfortunately didn't mention much about the other downsides though.
https://blog.cloudflare.com/kemtls-post-quantum-tls-without-...
About one third of traffic with Cloudflare is already using post-quantum encryption. https://x.com/bwesterb/status/1866459174697050145
Signatures still have to be upgraded, but that's more difficult. We're working on it. http://blog.cloudflare.com/pq-2024/#migrating-the-internet-t...