Explore quantum tomography and property testing in this comprehensive lecture from the Simons Institute. Delve into the design and estimation problems of quantum protocols, examining local asymptotic normality, compressed sensing, and projected least squares. Investigate testing under symmetry, Schur-Weyl duality, and the Clifford group. Learn about stochastic orthogonal transformations, stabilizer testing, and the exponential de Finetti theorem. Gain insights into quantum homeopathy and future perspectives in the field. Presented by David Gross from the University of Cologne, this talk offers a deep dive into advanced quantum concepts and their applications in testing and quantum PCPs.