Solve many quadratic equations with many unknowns at once. Hard in general.
The idea: Quadratic equations in many variables, such as x1·x3 + x2 + x4·x5 = 1, and dozens more like it, all at once. Checking a proposed solution is quick. Finding one is hard in general, for quantum computers too. Signatures use it by making the public key such a system: a valid signature is a solution for the message's hash, which only someone with a hidden shortcut can find.
Why it matters: The basis of some quantum-safe signatures.
On the map: solve random quadratic systems
Solve a system of random quadratic equations over a finite field. NP-hard in general and believed hard for quantum computers too.
It gives signatures (UOV, MAYO)[1, 2] with very small signatures but large public keys. It isn't known to give key exchange, which is why the family sits outside the public-key box.
If any of these breaks, this breaks too.
If this breaks, these break with it.