Deep learning (DL) models have provided state-of-the-art performance in various medical imaging benchmarking challenges, including the Brain Tumor Segmentation (BraTS) challenges. However, the task of focal pathology multi-compartment segmentation (e.g., tumor and lesion sub-regions) is particularly challenging, and potential errors hinder translating DL models into clinical workflows. Quantifying the reliability of DL model predictions in the form of uncertainties could enable clinical review of the most uncertain regions, thereby building trust and paving the way toward clinical translation. Several uncertainty estimation methods have recently been introduced for DL medical image segmentation tasks. Developing scores to evaluate and compare the performance of uncertainty measures will assist the end-user in making more informed decisions. In this study, we explore and evaluate a score developed during the BraTS 2019 and BraTS 2020 task on uncertainty quantification (QU-BraTS) and designed to assess and rank uncertainty estimates for brain tumor multi-compartment segmentation. This score (1) rewards uncertainty estimates that produce high confidence in correct assertions and those that assign low confidence levels at incorrect assertions, and (2) penalizes uncertainty measures that lead to a higher percentage of under-confident correct assertions. We further benchmark the segmentation uncertainties generated by 14 independent participating teams of QU-BraTS 2020, all of which also participated in the main BraTS segmentation task. Overall, our findings confirm the importance and complementary value that uncertainty estimates provide to segmentation algorithms, highlighting the need for uncertainty quantification in medical image analyses. Finally, in favor of transparency and reproducibility, our evaluation code is made publicly available at: this https URL.