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 constraints, which make it difficult to solve. In this work, we develop strengthened quadratic convex (QC) relaxations for ACOTS, where we tighten the relaxation with several new valid inequalities, including a novel kind of on/off cycle-based polynomial constraints by taking advantage of the network structure. We linearize the sum of on/off trilinear terms that appears in the relaxation with extreme-point representation and theoretically show its tightness. Also, we efficiently incorporate on/off cycle-based polynomial constraints using disjunctive programming based cutting planes. Combined with the optimization-based bound tightening algorithm, we obtain the tightest QC-based ACOTS relaxation to the best of our knowledge. Our extensive numerical experiments on medium-scale PGLib instances show significant improvement on relaxation bounds for many instances.