In this paper, we propose a new class of logical functions over residue ring of integers modulo p, where p is a prime. The magnitudes of the Chrestenson Spectra for this kind of functions, called as kth-order quasi-generalized Bent functions, take only two values—0 and a nonzero constant. By using the relationships between Chrestenson spectra and the autocorrelation functions for logical functions over ring Zp, we present some equivalent definitions of this kind of functions. In the end, we investigate the constructions of the kth-order quasi-generalized Bent functions, including the typical method and the recursive method from the technique of number theory.