As I know, isogeny-based cryptography and multivariate-based cryptography are known to have the smallest public keys and signature sizes, respectively. However, I don't have exact data one the sum. I've seen papers describing isogeny-based signatures with less-than-1000-bits public keys and signature sizes with sign and verification time of milliseconds. For instance, take a look at this one.
On the other hand, Rainbow, which is a multivariate-based signature scheme, has made it through to the third round of NIST post quantum contest, and thus, can be used with less concern about security. It says it has a 66-byte signature and is efficiently-computable.