I will give a brief introduction to the large deviation theorem for Toeplitz operators driven by proper dynamical systems. Then I will show several examples and discuss my recent results (joint with Jitomirskaya and Shi) about the large deviation theorem for general analytic \(k\)-frequency quasi-periodic operators on \(\mathbb{Z}^d\) for arbitrary \(k, d\). Finally, I will discuss several discrepancy problems arising from semi-algebraic sets and shifts/skew-shifts on tori.