Lattice-based cryptosystems for blockchain/ledger? Are there lattice-based cryptosystems based i.e., SIS (Short Integer Solutions) and LWE (Learning with Errors) blockchain solutions for a post quantum world?

Has the Unique Shortest Vector Problem (SVP) been solved?
Are there [...] blockchain solutions for a post quantum world?


Quantum Computing and related algorithms are no threat to the current protocol of Bitcoin (and most other blockchain systems I am familiar with) at all. Indeed, Shor's algorithm can solve the Discrete Logarithm Problem (DLP) efficiently, and a modified version exists for the elliptic curve analogue (ECDLP). Blockchain systems in their conventional form are based on


preimage and collision resistance of the employed hash functions for most functions, including transaction IDs, block IDs, and the Proof-of-Work (PoW) scheme in mining
hardness of DLP/ECDLP for security of transaction signatures and user addresses (so that private keys can not be found)


At the current state of research, most consider hash functions quantum resistant due to their lack of mathematical structure. As mentioned, (EC)DLP is not resistant.

However, if user addresses are never re-used then this is irrelevant, since an attacker will only learn a users clean (i.e. non-hashed) public key once a transaction is signed in the name of said user address/public key. As long as the hash function (e.g. SHA-256) is preimage-resistant, an attacker has nothing to apply Shor's algorithm on.
So there is no pressing need for alternatives yet. 

Nonetheless, lattice-based crypto is a proposal that comes up once in a while - to my knowledge no actual blockchain implementation exists as of today. As for the Shortest Vector Problem (SVP): the most efficient known algorithms are mentioned in the Wikipedia article.