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Proof of Work (PoW) protocols, originally proposed to circumvent DoS and email spam attacks, are now at the heart of the majority of recent cryptocurrencies. Current popular PoW protocols are based on hash puzzles. These puzzles are solved via a brute force search for a hash output with particular properties, such as a certain number of leading zeros. By considering the hash as a random function, and fixing a priori a sufficiently large search space, Grover’s search algorithm gives an asymptotic quadratic advantage to quantum machines over classical machines. In this paper, as a step towards a fuller understanding of post-quantum blockchains, we propose a PoW protocol for which quantum machines have a smaller asymptotic advantage. Specifically, for a lattice of rank n sampled from a particular class, our protocol provides as the PoW an instance of the Hermite Shortest Vector Problem (Hermite-SVP) in the Euclidean norm, to a small approximation factor. Asymptotically, the best known classical and quantum algorithms that directly solve SVP type problems are heuristic lattice sieves, which run in time 20.292n+o(n) and 2 0.265n+o(n) respectively. We discuss recent advances in SVP type problem solvers and give examples of where the impetus provided by a lattice-based PoW would help explore often complex optimization spaces.