Some zero-knowledge proofs needed a one-time ceremony whose secret must be destroyed by at least one participant.
The idea: These proof systems need public numbers built from a secret τ, so that provers can do math "at τ" without knowing it. Whoever knew τ could fake proofs, so it must be destroyed. A ceremony makes that easy to trust: each participant mixes in their own secret randomness, so the final τ is effectively the combination of all of them, and it stays unknown as long as a single participant threw their piece away.
Why it matters: If every participant colluded, fake proofs would be possible.
On the map: Powers of Tau + per-circuit ceremonies: ≥ 1 honest participant each
Groth16 needs secret randomness generated once per circuit, and PLONK needs it once for all circuits[1, 2]. Whoever knows that randomness (the "toxic waste") could forge proofs.
Ceremonies spread the secret across many participants: a "Powers of Tau" phase shared by many projects[3, 4], plus a per-circuit phase for Groth16. The result is safe if at least one participant in each phase destroyed their contribution.
If any of these breaks, this breaks too.
If this breaks, these break with it.