I took some rough notes to whittle down the verbiage.
This proposal is to introduce PQ certificates in WebPKI such as for certificate authorities.
Problem is PQ signatures are large. If certificate chain is small that could be acceptable, but if the chain is large, then it can be expensive in terms of bandwidth and computation during TLS handshake. That is the exchange sends many certificates which embed a signature and a large (PQ) public key.
Merkle Tree Certificates ensures that an up to date client only needs 1 signature, 1 public key, 1 merkle tree witness.
Looking at an MTC generated certificate they've replaced the traditional signing algorithm and signature with a witness.
That means all a client needs is a signed merkle root which comes from an expanding Merkle Tree signed by the MTCA (Merkle Tree CA), which is delivered somehow out of band.
So basically TLS client receives certificate containing new signature algorithm which embeds a witness instead of a signature, a root (not sure if just a hash or a signed hash, I think the former). Client will get the signed roots out of band, which can be pre-verified, which means verifying the witness is simply doing a check on the witness.
Edit: My question: is this really a concern that needs to be addressed? PQ for TLS key exchange addresses a looming threat of HNDL (Harvest Now Decrypt Later). I don't see why we need to address making WebPKI use PQ signatures, at least for awhile now.