Public key cryptography, used in daily life today, is vulnerable to quantum computer attacks. Setting up trapdoor systems, not using factoring and discrete log, is a first step towards designing quantum resilient systems. This talk will outline relevance of the problem of solving multivariate quadratic systems to designing such quantum resilient systems. Attacks on such cryptosystems will also be discussed. Some designs from NIST Post-Quantum Cryptography Competition will be outlined.