How efficiently can we check a computation?

In computer science we often ask: given a problem, how efficiently can we compute a solution? My work takes a different perspective, asking: if someone claims to have already computed a solution, how efficiently can we check it’s correct? This question has deep connections with many areas of theoretical computer science, including cryptography, complexity theory and quantum computing; and, more recently, has had significant practical impact in decentralised systems. In this talk I will focus on two aspects of my work in this area: first, on designing concretely efficient checking protocols; and second, on ensuring the integrity of efficient checking against quantum attackers.