Quadratic programming (QP) underpins real-time robotics by enabling efficient, constrained optimization in state estimation, motion planning, and control. In legged locomotion and manipulation, essential modules like inverse dynamics, Model Predictive Control (MPC), and Whole-Body Control (WBC) are inherently QP-based, demanding reliable solutions amid tight timing, energy, and computational resources on embedded platforms. This paper presents a comprehensive analysis and benchmarking study of QP solvers for legged robotics. We begin by formulating the standard convex QP and classify solvers into principal algorithmic approaches: interior-point methods, active-set strategies, operator-splitting schemes, and augmented Lagrangian/proximal approaches, while also discussing solver code generation for fixed-structure QPs. Each solver is examined in terms of algorithmic structure, computational characteristics, and its ability to exploit problem structure and warm-starting. Performance is reviewed using publicly available benchmarks, with a focus on metrics such as computation time, constraint satisfaction, and robustness under perturbations. Unified comparison tables yield practical guidance for solver selection, underscoring trade-offs in speed, accuracy, and energy efficiency. Our findings emphasize the synergy between solvers, tasks, and hardware -- e.g., sparse structured IPMs for long-horizon MPC and dense active-set for high-frequency WBC to advance agile, autonomous legged systems, with emerging trends toward ill-conditioned, conic, and code-generated deployments.

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