Experiments on single colloidal quantum dots (QDs) have revealed temporal fluctuations in the emission efficiency of the single-exciton state. These fluctuations, often termed "blinking", are caused by opening/closing of charge-carrier traps and/or charging/discharging of the QD. In the regime of strong optical excitation, multiexciton states are formed. The emission efficiencies of multiexcitons are lower because of Auger processes, but a quantitative characterization is challenging. Here, we quantify fluctuations of the biexciton efficiency for single CdSe/CdS/ZnS core-shell QDs. We find that the biexciton efficiency "blinks" significantly. The additional electron due to charging of a QD accelerates Auger recombination by a factor of 2 compared to the neutral biexciton, while opening/closing of a charge-carrier trap leads to an increase of the nonradiative recombination rate by a factor of 4. To understand the fast rate of trap-assisted recombination, we propose a revised model for trap-assisted recombination based on reversible trapping. Finally, we discuss the implications of biexciton blinking for lasing applications.