Boaz Barak

1. It confirms a 1986 conjecture of H̊astad (also raised by and discussed by many others) that the polynomial hierarchy is infinite with respect to a random oracle. (We’ve already had strong reasons to believe H̊astad’s conjecture, since Book showed in 1994 that it’s true unless the polynomial hierarchy collapses.) The prior best result was by O’Donnell and Wimmer in 2007, who showed that Σ3 6⊆ Σ2 w.r.t. a random oracle. However, they did not state their result in this language, and it’s unclear if they were aware of this connection. (Aaronson 2010, showed the (incomparable?) result that Π2 6⊆ PNP with respect to a random oracle, and also realized the connections between these questions and questions on the analysis of Boolean functions, though not the relation to the work of O’Donnell and Wimmer.)