The Dempster-Shafer (DS) belief theory constitutes a powerful framework for modeling and reasoning with a wide variety of uncertainties due to its greater expressiveness and flexibility. As in the Bayesian probability theory, the DS theoretic (DST) conditional plays a pivotal role in DST strategies for evidence updating and fusion. However, a major limitation in employing the DST framework in practical implementations is the absence of an efficient and feasible computational framework to overcome the prohibitive computational burden DST operations entail. The work in this article addresses the pressing need for efficient DST conditional computation via the novel computational model DS-Conditional-All. It requires significantly less time and space complexity for computing the Dempster's conditional and the Fagin-Halpern conditional, the two most widely utilized DST conditional strategies. It also provides deeper insight into the DST conditional itself, and thus acts as a valuable tool for visualizing and analyzing the conditional computation. We provide a thorough analysis and experimental validation of the utility, efficiency, and implementation of the proposed data structure and algorithms. A new computational library, which we refer to as DS-Conditional-One and DS-Conditional-All (DS-COCA), is developed and harnessed in the simulations.