Achieving universal and scalable quantum computing with reliably low error rates, despite the presence of unreliable circuit components, requires fault-tolerant quantum error correction. In general, quantum error correction imposes a significant overhead on the computation, motivating exploration of opportunities for optimization. Flag fault tolerance protocols have emerged as important schemes to realize fault tolerance experiments in the near term, because of their low qubit overhead, and absence of strict requirement for elaborate ancillary state preparation, relative to traditional schemes. However, the existing fast-reset, single-flag protocols for small codes generally employ a measurement of all stabilizer generators with unflagged circuits to distinguish a limited set of errors via the syndrome, leading to high circuit depth. In addition, the flagged measurement outcomes play a limited role in differentiating these errors. This motivates the possibility of reducing the circuit depth fault-tolerantly in flag-based syndrome extraction circuits. In this thesis, flag protocols with significantly reduced number of stabilizer measurements are constructed for the [[5,1,3]] code and the Steane code. The new protocols are divided into two classes. In the first class, the reduction is achieved by a dynamic choice of unflagged stabilizer measurements, based on past syndromes, and the utilization of the complete stabilizer group, to distinguish restricted sets of errors signalled by respective flagged measurements. In the second class, the reduction is achieved by measuring three high-weight flagged stabilizers, with the capability to detect a single input error, for the Steane code. The reduced stabilizer sequences are methodically constructed to yield unique and nontrivial syndromes for the relevant error set. This ensures that the fundamental condition of errors being detectable and distinguishable, which is the principal factor for the existing flag protocols to be fault-tolerant, is preserved. Pseudothresholds competitive with the existing flag protocols are established via Monte Carlo simulations under an error model consisting of two-qubit gate depolarizing errors, state preparation errors and measurement errors. Additionally, computer search programs are developed to obtain analogous reduced stabilizer sequences for both classes. These programs are also employed to assist in identifying certain mathematical properties of the high-weight Steane code stabilizers which can detect a single input error: namely, these stabilizers belong to different cosets of the X-stabilizer subgroup, and arise from 8-element subgroups within the stabilizer group. Furthermore, examples of such stabilizer sequences are constructed for few other codes. This thesis highlights the potential of employing parity measurements from the complete stabilizer group and extending beyond conventional adaptive measurements to improve the resource efficiency of fault-tolerant quantum error correction.