We give improved inapproximability results for some minimization problems in the second level of the Polynomial-Time Hierarchy. Extending previous work by Umans [Uma99], we show that several variants of DNF minimization are p2-hard to approximate to within factors of n13− and n12− (where the previous results achieved n14− ), for arbitrarily small constant 0">0. For one problem shown to be inapproximable to within n12− , we give a matching O(n12) -approximation algorithm, running in randomized polynomial time with access to an NP oracle, which shows that this result is tight assuming the PH doesn't collapse.