Common misconception: ZKP is not always about 'circuits'. Actually, what we are proving are 'relations'. It just happens that most of the early deployed SNARKs were for relations that were best expressed by (arithmetic) circuits. So, the name caught on.

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We're entering a phase where provers for special relations are becoming commonplace. E.g. existence of a value in a table is a good example of such a relation. Evaluations of a polynomial over {0,1}^n is another.

What's tricky is mapping the logic (i.e. the program) we want to prove to a relation with an efficient prover. For the above two examples we have "Lookup arguments" (aptly/confusingly named to prove lookup relations) and sumcheck.