Recent breakthroughs in quantum technology, suchas quantum computing, communication, and metrology have given rise to questions related to quantum measurements which can be formulated in the language of mathematical statistics. Since quantum mechanics is fundamentally non-commutative in nature, statistical inference for these problems also involves dealing with such non-commutative structures. Moreover, inference in quantum statistics based on the laws of quantum probability deviates from inference in classical statistics, and the results often turn out to be different in a non-trivial way. In classical statistics a fundamental paradigm is approximating complicated experiments (families of laws, or models) by simpler ones. In particular, one establishes asymptotic equivalence between i.i.d. models indexed by a local parameter and a Gaussian shift model (with the shift given by the same local parameter). This approximation is called local asymptotic normality (LAN) and allows one to construct an estimator from a procedure in the Gaussian model with similar risk bounds. Local asymptotic equivalence can also been established between quantum i.i.d. models and quantum Gaussian models. In this thesis, we explore quantum statistical inference through the lens of local asymptotic equivalence and establish quantum counterparts of several results in classical statistics.