With the surging interest in quantum and hybrid classical-quantum systems, ensuring the privacy of generated quantum data is essential. Quantum differential privacy (QDP) has been introduced to ensure privacy for quantum states. However, the versatility of QDP is limited. We introduce a flexible privacy framework for quantum systems termed as Quantum PP (QPP). We show that QPP is captured exactly by an information-spectrum divergence, endowing the latter with its first operational interpretation. Then this divergence is used to study properties of QPP mechanisms, showcasing distinctions that arise in the quantum setting compared to the classical setting. We introduce the first algorithms that can be used to audit for privacy given a mechanism, which facilitates new approaches with the access to quantum devices. We analyse how the QPP framework provides better privacy-utility tradeoffs with the flexibility to incorporate application-specific criteria.To provide further insights on QPP, we comprehensively study measured hockey-stick divergences that find operational interpretation in the QPP framework as optimal privacy parameters. Studying statistical problems under privacy constraints is vital for understanding the price that we have to pay to ensure privacy. To this end, the contraction of statistical measures and divergences under privacy constraintsis an important technical tool. However, in the quantum setting, this area of research is largely unexplored even for fundamental statistical tasks. We characterize the contraction of quantum divergences under a local variant of quantum privacy, which enables this field of study. We completely characterize the privatized contraction coefficient of the trace distance, among others. Next, we utilize the information-theoretic tools developed to study statistical tasks under privacy constraints. To this end, we characterize the cost of privacy in quantum hypothesis testing and learning expectation of observables while ensuring privacy of quantum states. In sum, this thesis lays a theoretical foundation for ensuring the privacy of quantum data by introducing flexible privacy frameworks tailored to quantum systems. It also evaluates the tradeoffs involved in maintaining privacy during the testing and learning of properties of quantum data, offering information-theoretic tools to study statistical tasks under privacy constraints.