The key is the "any arbitrary program". With zk-snarks, I can, theoretically, run some complex analysis of some data -- imagine something that requires million

The key is the "any arbitrary program". With zk-snarks, I can, theoretically, run some complex analysis of some data -- imagine something that requires millions of compute hours -- and provide a receiver (a) the answer, and (b) a compact hash-like proof that (a) is correct *without* receiver having to re-run the calculation to trust the answer.