The purpose of this paper is to extend various properties of positive-definite functions from groups to quantum groups. The authors start from the notions of positive definiteness on locally compact quantum groups introduced in [M. Daws, Int. J. Math. 23, No. 12, Paper No. 1250132, 23 p. (2012; Zbl 1282.43002)] and [M. Daws and P. Salmi, J. Funct. Anal. 264, No. 7, 1525–1546 (2013; Zbl 1320.46056)] and prove in the settings of locally compact quantum groups exciting results. “Among these are theorems on “square roots” of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups.”