Existing temporal community search suffers from two defects: (i) they ignore the temporal proximity between the query vertex $q$ and other vertices but simply require the result to include $q$. Thus, they find many temporal irrelevant vertices (these vertices are called \emph{query-drifted vertices}) to $q$ for satisfying their cohesiveness, resulting in $q$ being marginalized; (ii) their methods are NP-hard, incurring high costs for exact solutions or compromised qualities for approximate/heuristic algorithms. Inspired by these, we propose a novel problem named \emph{query-centered} temporal community search to circumvent \emph{query-drifted vertices}. Specifically, we first present a novel concept of Time-Constrained Personalized PageRank to characterize the temporal proximity between $q$ and other vertices. Then, we introduce a model called $β$-temporal proximity core, which can combine temporal proximity and structural cohesiveness. Subsequently, our problem is formulated as an optimization task that finds a $β$-temporal proximity core with the largest $β$. 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.