Simultaneous inference allows for the exploration of data while deciding on criteria for proclaiming discoveries. It was recently proved that all admissible post-hoc inference methods for true discoveries must employ closed testing. In this paper, we investigate efficient closed testing with local tests of a special form: thresholding a function of sums of test scores for the individual hypotheses. Under this special design, we propose a new statistic that quantifies the cost of multiplicity adjustments, and we develop fast (mostly linear-time) algorithms for post-hoc inference. Paired with recent advances in global null tests based on generalized means, our work instantiates a series of simultaneous inference methods that can handle many dependence structures and signal compositions. We provide guidance on the method choices via theoretical investigation of the conservativeness and sensitivity for different local tests, as well as simulations that find analogous behavior for local tests and full closed testing.