Most Compact Secure Post-Quantum Signatures and Public Keys? Which post-quantum public key and signature system includes the most compact reusable keypairs?  E.g. a private key which produces practically unlimited signatures without losing the security provided initially like RSA as opposed to something like hash-based Merkle tree driven signatures with a limited number of uses?
First of all, you have to say bye to the assumption that you get something like RSA or even ECDSA. Sizes for post-quantum candidates are larger than for classical schemes. Period. There exist schemes with small public keys or signatures but never both. The point is that you can often shift information around between public key and signature but the sum remains the same or gets even worse. 

Generally, you got three areas that you can pick from: Hash-based, lattice-based, or multi-variate signatures. Code-based and isogeny-based do not really have practical proposals, yet. Each of these areas has its advantages and disadvantages. 

Hash-based gives you the most reliable security guarantees, small public keys and moderate size signatures. For small, fast signatures you require to maintain a state (XMSS / LM-HSS), for stateless schemes (SPHINCS, SPHINCS+, Gravity-SPHINCS) signatures get to the order of 10th of kilobytes. 

Lattice-based crypto currently provides you with the best sizes. If the current parameters turn out to be secure you can get away with 2.5 kilobyte signatures and 1.5 kilobyte public keys (Dilithium). However, the cryptanalysis of lattice problems is still on the move. It would not be totally surprising if someone finds an algorithm that is an order of magnitude faster than current attacks (this does not mean that the schemes would be broken but that parameters would have to be adjusted). 

Finally, for MQ or multivariate signatures you got two choices: You can go with schemes like UOV or Rainbow (careful: patents) which offer extremely  short signatures but somewhat large public keys (I do not know exact numbers  as all I found are numbers for classical attacks and 80bit security...). If you further add cyclic structure to the keys, public key sizes go down. However, these constructions are heuristic and we only know that they were not yet broken with known attacks. Especially, their security is not only linked to the MQ-problem but also to far less understood problems like IP and MinRank. Alternatively, you can go with something like MQDSS which has a security reduction from MQ. However, then you are at slightly larger sizes than SPHINCS.