Abstract: Quantum intersection numbers were introduced through a natural quantization of the KdV hierarchy in a work of Buryak, Dubrovin, Guere, and Rossi. Because of the Kontsevich-Witten theorem, a part of the quantum intersection numbers coincides with the classical intersection numbers of psi-classes on the moduli spaces of stable algebraic curves. I will talk about our joint work in progress with Xavier Blot, where we relate the quantum intersection numbers to the stationary relative Gromov-Witten invariants of the Riemann sphere, with an insertion of a Hodge class. Using the Okounkov-Pandharipande approach to such invariants (with the trivial Hodge class) through the infinite wedge formalism, we then give a short proof of an explicit formula for the ``purely quantum'' part of the quantum intersection numbers, found before by Xavier, which in particular relates these numbers to the one-part double Hurwitz numbers.