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][1].


  [1]: http://www.site.uottawa.ca/~cadams/papers/prepro/paper_30.pdf