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 \Sigma_2^p-hard to approximate to within factors of n^{1/3-\epsilon} and n^{1/2-\epsilon} (where the previous results achieved n^{1/4 - \epsilon}), for arbitrarily small constant \epsilon > 0. For one problem shown to be inapproximable to within n^{1/2 - \epsilon}, we give a matching O(n^{1/2})-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.