Strongly unforgeable digital signatures exist from one-way function, so they are indeed a Minicrypt assumption, even though most efficient construction use public key cryptography.
For succinct zero-knowledge, Kilian's protocol is a succinct zero-knowledge argument for all of NP assuming only collision-resistant hash functions, which is also typically seen as a Minicrypt assumption. Whether OWF could suffice for this is a long-standing open problem.
Using the Fiat-Shamir heuristic, you can turn it into a transparent SNARK. If you want something more concretely efficient, try Ligero, for example.
If you don't want to use the Fiat-Shamir heuristic or to assume random oracles, then indeed we have an unsatisfying situation:
- SNARKs can be constructed unconditionally from a random oracle, so they are very much a Minicrypt-style assumption, yet
- We don't have any provably secure constructions of them from any standard Minicrypt assumption like OWF, CRHF or others.
Note, however, that there's nothing specific to succinct arguments or tu trusted setups for that: the same applies to non-interactive zero-knowledge proofs in general (without succinctness, and even with trusted setup). Getting NIZKs from Minicrypt assumptions is a long-standing, probably very hard open problem. Indeed, until recently we did not even have NIZKs from most public-key-style assumptions, such as LWE or DDH! (For the former, we now have a construction due to Peikert and Shiehan, and for the latter, Jain and Jin recently built one from subexponential DDH)
