No post-quantum roadmap, improvement proposal or research programme specific to Dogecoin was found at the time of writing. source
1 Signature scheme 0/10, weighted 30%
Band 0: Classical signatures only on mainnet, with nothing post-quantum protecting live funds. Short-lived devnets and one-off benchmarks sit here: they demonstrate research rather than something a user can hold keys on.
Dogecoin uses ECDSA on secp256k1, inherited from the Bitcoin codebase. Nothing post-quantum protects DOGE on mainnet. source