Core/shell semiconductor nanocrystals can be designed to afford type I, quasi-type II, or type II energy-band alignments, differing in the nature of the carrier-relaxation processes. In this study, we synthesized CdTe/CdS core/shell quantum dots (QDs) from four differently sized CdTe core QDs by a hydrothermal method, and we examined their exciton-relaxation dynamics by time-resolved luminescence and transient absorption (TA) spectroscopies. We performed core/shell-selective excitation experiments on large CdTe/CdS core/shell QDs with tetrahedral structures. The relative bleach TA intensities at the 1S state against the lowest exciton state of the CdS shell are different for the core and the shell excitations. We interpret this phenomenon in terms of a small potential barrier in the conduction band, induced by grain boundaries at the core/shell interface of CdTe/CdS core/shell QDs. The evolution of the band alignment from type I to quasi-type II is confirmed by increased luminescence lifetimes, hole localization in the core, and electron delocalization throughout the QD. Moreover, the biexciton Auger recombination lifetime (τ Auger ) of CdTe/CdS core/shell QDs represents a luminescence wavelength dependence that is similar to that of CdTe QDs, which is further support for the quasi-type II band alignment. In the present of potential energy barrier, τ Auger of the large CdTe/CdS core/shell QDs becomes elongated. This study offers a useful guideline for determining the energy-band alignment of semiconductor nanoheterostructures not only according to luminescence lifetimes and exciton-relaxation kinetics but also τ Auger .