A quantum computer is a computer that exploits quantum mechanical phenomena: for this reason, it is able to perform tasks that the classical computer cannot do, including solving the known (and therefore currently unsolvable) problems on which cryptographic systems are currently based. Post-quantum cryptography deals with studying new quantum-resistant problem on which to rely in order to be able to run secure signature protocols. In this context, we can observe that the digital signatures can be based on different problems, more or less complex: the following thesis arise from the interest in analysing one of these, the multivariate quadratic problem, in all its facets, from its definition to its use in zero-knowledge protocols, which in turn can fall into different types.