As an infinite dimensional generalization of the usual Laplacian on an Euclidean space the so-called Lévy Laplacian $$
\Delta_L=\lim_{N\to\infty}\frac{1}{N}\sum^N_{n=1}\frac{\partial^2}{\partial x^2_n}
$$
was introduced and studied by Lévy in his famous books, and has been investigated from various aspects by many authors. In recent years the Lévy Laplacian has afforded us much interest for its newly discovered relations with certain stochastic processes, Yang-Mills equations, Gross Laplacian, infinite dimensional rotation group, quadratic quantum white noise, Poisson noise functionals, and further relevant questions. In this talk, we introduce a non-commutative generalization of the Lévy Laplacian, called the quantum Lévy Laplacian, acting on white noise operators.