What's Up with Downward Collapse: Using the Easy-Hard Technique to Link Boolean and Polynomial Hierarchy Collapses

During the past decade, nine papers have obtained increasingly strong consequences from the assumption that boolean or bounded-query hierarchies collapse. The final four papers of this nine-paper progression actually achieve downward collapse---that is, they show that high-level collapses induce collapses at (what beforehand were thought to be) lower complexity levels. For example, for each $k\geq 2$ it is now known that if $\psigkone=\psigktwo$ then $\ph=\sigmak$. This article surveys the history, the results, and the technique---the so-called easy-hard method---of these nine papers.