Specifically, say I connect via TLS to a server that uses FS and exchange some information that will be valuable in 150 years. If the entire conversation is recorded at the wire, does quantum computing break this promise? (or have I misunderstood FS completely?)
According to https://crypto.stackexchange.com/questions/5610/diffie-hellm..., Diffie-Hellman is not post-quantum secure, but https://en.wikipedia.org/wiki/Supersingular_isogeny_key_exch... is a DH alternative that is.
Here’s an overview of the performance from a patch by Armando Faz Hernandez: https://github.com/cloudflare/p751sidh/pull/2
If you can break the ECDH, you can figure out the key that was calculated. However, if other unknown data (e.g. past ECDH outputs) is hashed together to generate the key (like Signal does, as a very rough description), it's only post-quantum secure if the adversary ever misses a single message.
Other than weird saved-by-the-bell corner cases, the answer is "No".
https://tonyarcieri.com/imperfect-forward-secrecy-the-coming...
No, forward secrecy doesn't apply if the underlying public-key cryptography gets broken (such as by a quantum computer).