Temporal knowledge graph question answering (TKGQA), which utilizes temporal knowledge graphs (TKGs) to answer natural language questions, has attracted increasing attention in recent years. Real-life applications of TKGQA tend to be complex in temporal granularity, i.e., the questions may concern mixed temporal granularities (e.g., both day and month). Nevertheless, multi-granularity temporal knowledge graph question answering (Multi-granularity TKGQA) remains underexplored. Existing methods are mainly designed for simple multi-granularity temporal questions that can be answered by a single TKG fact, struggling with multi-granularity temporal questions requiring multi-fact reasoning and implicit temporal inference. This paper proposes a comprehensive embedding-based framework for multi-granularity complex TKGQA, namely MCTQA, which consists of three core modules. First, a question representation module encodes the question into semantic embeddings. Second, a graph structure-aware module incorporates TKG structural information to support multi-fact reasoning by treating entities in question as “bridges”. Third, a multi-granularity temporal fusion module identifies implicit temporal signals in the question and enhances the question representation by generating granularity-specific time embeddings using a novel time anchor-based time embedding generation method. Experimental results prove that MCTQA significantly outperforms strong baselines on the challenging MULTITQ dataset, achieving an absolute improvement of up to 6% in overall Hits@1. The code is available at https://github.com/Leserein-yjh/MCTQA .