Background
Quantum-secure (asymmetric) signature schemes have 1kB+ signature sizes. The public keys also tend to be large. Wikipedia has a size comparision table for various signature schemes.
Below we combine sequential PoW with a symmetric signature scheme to yield a succinct quantum-secure asymmetric signature scheme. The private and public keys are 32 bytes, and the signature is 64 bytes.
Construction
Let e be 32 bytes of entropy acting as a private key. Let H(e) be the corresponding public key. Let t be a transaction with sender H(e).
The signer first produces a proof p of sequential work for [t, e]. For example p = H^n([t, e]) for some pre-specified large n. The signature for t is then [e, p]. Only the first (as recorded by the blockchain) signed transaction revealing e with a correct p counts as having a valid signature.
The PoW parameter n is a security parameter large enough to prevent front-running of the symmetric private key e on a different transaction t'.