We study generic properties of high-dimensional quantum states. Specifically, for a random state on H=C^d otimes C^d obtained by partial tracing a random pure state on H otimes C^s, we consider the problem whether it is typically separable or typically entangled. We show that a threshold occurs when the ancilla dimension s is of order roughly d^3. Our approach allows to similarly analyze other properties such as for example positive partial transpose (PPT). Mathematically, each problem reduces to studying properties (albeit somewhat exotic) of high-dimensional complex Wishart matrices. The arguments rely on high-dimensional probability, classical convexity, random matrices, and geometry of Banach spaces. Based on joint work with G. Aubrun and D. Ye.