We investigate the effects of laser pulse shape on strong-field quantum electrodynamics (QED) processes during the collision between a relativistic electron beam and an intense laser pulse. The interplay between high-energy photon emission and two pair production processes, i.e. nonlinear Breit–Wheeler (BW) and Trident, was investigated using particle-in-cell simulations. We found that the temporal evolution of these two processes could be controlled by using laser pulses with different degrees of asymmetry. The temporal envelope of the laser pulse can significantly affect the number of pairs coming from the Trident process, while the nonlinear BW process is less sensitive to it. This study shows that the two QED processes can be examined with state-of-the-art petawatt lasers and the discrimination of the two pair creation processes is feasible by adjusting the temporal asymmetry of the colliding laser pulse.