ZK FAQ: What's a trusted setup? What's a Structured Reference String? What's toxic waste? posted 4 weeks ago
In proof systems, provers and the verifiers rely on a common set of parameters, sometimes referred to as the common reference string (CRS).
In some proof systems (for example, the ones that rely on pairings) a dangerous setup phase produces these common parameters. Dangerous because it generates random values, encrypts them, and then must get rid of the random values so that no one can ever find out about them. The reason is that knowing these values would allow anyone to forge invalid proofs. Invalid proofs that verifiers would accept. Such values are sometimes referred to as toxic waste, and due to the fact that the individuals performing the setup have behave honestly, we call the setup a trusted setup.
By the way, since this common set of parameters has some hidden structure to it, it is usually referred to as structured reference string (SRS).
In the past, ceremonies (called powers of tau ceremonies) have been conducted where multiple participants collaborate to produce the SRS. Using cryptographic constructions called multi-party computations (MPC), the protocol is secure as long as one of the participant behaves honestly (and destroys the random values they generated as part of the ceremony).
It seems to be accepted in the community that such ceremonies are a pain to run. When mistakes happen, new ceremonies have to take place, which is what infamously happened to Zcash.