At this point it seems unlikely that a breakthrough in crypto could kill blockchains without killing nearly all of modern cryptography. For instance we can build blockchains which are secure even against quantum computers. We can build blockchains that make no number theoretic assumptions using only secure hash functions. Any technique which can break all techniques we have of inventing hard to solve problems (i.e. cryptography), would have a massive impact on technology.

A breakthrough in crypto that would destroy blockchains is much more likely to be a technology which is significantly better than a blockchain for what we want to do with blockchains. That would also be a exciting result.