Are there digital signatures for which, given two documents signed by the same key, one could derive the key?

With such one-time signatures, one may be able to design a cryptocurrency based on proof-of-stake instead of proof-of-work. To disincentivise miners from cheating by mining several blocks, one could impose that miners sign with a one-time digital signature the blocks they mine. That way, if a miner did mine two different blocks with the same key, the key would be exposed and they would lose the funds associated with that key.