In the 1995 paper entitled "Noncommutative symmetric functions," Gelfand et al. defined several noncommutative symmetric function analogues for well-known symmetric function bases, including two distinct types of power sum bases. This paper explores the combinatorial properties of their duals, two distinct quasisymmetric power sum bases. In particular, we show that they refine the classical symmetric power sum basis, and give transition matrices to other well-understood bases, as well as explicit formulas for products of quasisymmetric power sums.