The Alternating Current Optimal Transmission Switching (ACOTS) problem incorporates line switching decisions into the fundamental AC optimal power flow (ACOPF) problem. The advantages of the ACOTS problem are well-known in terms of reducing the operational cost and improving system reliability. ACOTS optimization models contain discrete variables and nonlinear, non-convex structures, which make it difficult to solve. We derive strengthened quadratic convex (QC) relaxations for ACOTS by combining several methodologies recently developed in the ACOPF literature. First, we relax the ACOTS model with the on/off QC relaxation, which has been empirically observed to be both tight and computationally efficient in approximating the ACOPF problem. Further, we tighten this relaxation by using strong linearization with extreme-point representation, and by adding several types of new valid inequalities. In particular, we derive a novel kind of on/off cycle-based polynomial constraints, by taking advantage of the network structure. Those constraints are linearized using convex-hull representations and implemented in an efficient branch-and-cut framework. We also tighten the relaxation using the optimization-based bound tightening algorithm. Our extensive numerical experiments on medium-scale PGLib instances show that, compared with the state-of-the-art formulations, our strengthening techniques are able to improve the quality of ACOTS relaxations on many of the PGLib instances, with some being substantial improvements.