One important weakness of zkSnarks is that it requires a trusted setup, for example [1]. A new alternative is called zk-STARK [2], which doesn't require the trusted setup, and is post-quantum secure. However, it significantly increases the size of the proof (around ~50KB). In general, hash-based post-quantum algorithms require bigger size and it would be interesting to watch the progress made in this regard.
[1]https://filecoin.io/blog/participate-in-our-trusted-setup-ce...