ZK-Rollup
A rollup that posts a cryptographic proof with each batch showing that the new state was computed correctly.
Instead of waiting to see whether anyone objects, the operator proves the batch, and a verifier contract on the base chain checks the proof before accepting the new state, so withdrawals can be honored as soon as a proof is verified. The cost moves to the prover, which needs substantial hardware and is a centralized component in most current systems, and the proving circuits themselves are complex enough that bugs in them are a genuine risk category. A zkEVM aims to prove execution of ordinary EVM bytecode so existing contracts work unchanged, with implementations differing in how closely they match Ethereum's behavior. zkSync Era, Starknet, Polygon zkEVM, Scroll and Linea are examples, and despite the name most of them offer no transaction privacy.
In practice
Because the base chain verifies a proof rather than waiting out a challenge period, a validity rollup can process a withdrawal without the multi-day delay optimistic designs need.
The common misunderstanding
A zk-rollup is usually not private; the zero-knowledge machinery is used for compact verification, and its transactions remain publicly visible.