Well, I really think that these guys have brilliant ideas, however, I am not yet sure how this thing is supposed to work in practise. But I may just not understand it fully (yet).
The way I understand it right now is that their approach makes it pretty easy to verify the soundness of transactions without the need to have a full copy of the Blockchain. So basically, you could just download the latest state and verify the zk-SNARKs which prove that there exists a valid transaction chain up to the genesis block that would result in that exact state. Something, you would need the full copy of the Blockchain for in more traditional systems; how else could you make sure that a transaction of mine is actually valid (in terms of that I have really received the coins I'm spending in the past, and not just claiming so).

So in that sense, this approach is pretty cool. The biggest question mark is though ... if you are just checking the zkSNARK to see whether there exists a transaction chain all the way back to the genesis block, you are exposed to all sorts of attacks: the simplest one is most likely to just create a private side chain where you create a state that you want to spoof to someone else, and somehow submit this state along with perfectly valid zk-SNARKs to your victim. How in this world can he know that state is invalid and that there exists a different state with a "greater cumulative difficulty" containing the same coins spent to someone else (longest chain approach)? I cannot think of any way, but like I said, probably I am just not understanding it correctly

I think to mitigate this issue you would at least need to downlod all block headers to verify your „state“ is actually in the longest chain which wouldn't make it "succinct" anymore.