This paper introduces UnSeenTimeQA, a novel data contamination-free time-sensitive question-answering (TSQA) benchmark. It differs from existing TSQA benchmarks by avoiding web-searchable queries grounded in the real world. We present a series of time-sensitive event scenarios based on synthetically generated facts. It requires large language models (LLMs) to engage in genuine temporal reasoning without depending on the factual knowledge acquired during the pre-training phase. Our data generation framework enables on-demand generation of new samples, mitigating the risk of data leakage. We designed three types of time-sensitive questions to test LLMs’ temporal reasoning abilities over sequential and parallel event occurrences. Our evaluation of five LLMs on synthetic fact-based TSQA reveals mixed results: while they perform well on simpler subsets, their overall performance remains inferior as compared to real world fact-based TSQA. Error analysis indicates that LLMs face difficulties in reasoning over long-range event dependencies and parallel events.