Temporal community search is an important task in graph analysis, which has been widely used in many practical applications. However, existing methods suffer from two major defects: (i) they only require that the target result contains the query vertexq, leading to the temporal proximity betweenqand other vertices being ignored. Thus, they may find many temporal irrelevant vertices (these vertices are called query-drifted vertices) concerningqfor satisfying their objective functions; (ii) their methods are NP-hard, incurring high costs for exact solutions or compromised qualities for approximate/heuristic algorithms. In this paper, we propose a new problem named query-centered temporal community search to overcome these limitations. Specifically, we first present a novel concept of Time-Constrained Personalized PageRank to characterize the temporal proximity betweenqand other vertices. Then, we introduce a model calledβ-temporal proximity core, which can seamlessly combine temporal proximity and structural cohesiveness. Subsequently, our problem is formulated as an optimization task that finds aβ-temporal proximity core with the largestβ.We theoretically prove that our problem can circumvent these query-drifted vertices. To solve our problem, we first devise an exact and near-linear time greedy removing algorithm that iteratively removes unpromising vertices. To improve efficiency, we then design an approximate two-stage local search algorithm with bound-based pruning techniques. Finally, extensive experiments on eight real-life datasets and nine competitors show the superiority of the proposed solutions.