Side channels pose a significant threat to the confidentiality of software systems. Such vulnerabilities are challenging to detect and evaluate because they arise from non-functional properties of software such as execution times and require reasoning on multiple execution traces. Recently, noninterference notions have been adapted in static analysis, symbolic execution, and greybox fuzzing techniques. However, noninterference is a strict notion and may reject security even if the strength of information leaks are weak. A quantitative notion of security allows for the relaxation of noninterference and tolerates small (unavoidable) leaks. Despite progress in recent years, the existing quantitative approaches have scalability limitations in practice. In this work, we present QFuzz, a greybox fuzzing technique to quantitatively evaluate the strength of side channels with a focus on min entropy. Min entropy is a measure based on the number of distinguishable observations (partitions) to assess the resulting threat from an attacker who tries to compromise secrets in one try. We develop a novel greybox fuzzing equipped with two partitioning algorithms that try to maximize the number of distinguishable observations and the cost differences between them. We evaluate QFuzz on a large set of benchmarks from existing work and real-world libraries (with a total of 70 subjects). QFuzz compares favorably to three state-of-the-art detection techniques. QFuzz provides quantitative information about leaks beyond the capabilities of all three techniques. Crucially, we compare QFuzz to a state-of-the-art quantification tool and find that QFuzz significantly outperforms the tool in scalability while maintaining similar precision. Overall, we find that our approach scales well for real-world applications and provides useful information to evaluate resulting threats. Additionally, QFuzz identifies a zero-d...