Abstract: Quantum Loewner evolution (QLE), introduced by Miller and Sheffield, is a continuum growth process on Liouville quantum gravity surfaces, proposed as the scaling limit of growth models on random planar maps (e.g., diffusion-limited aggregation on spanning-tree-weighted random planar maps). We construct a variant of QLE whose time parametrization is the continuum analog of the intrinsic time parametrization of discrete growth models. I will describe the motivation from discrete growth processes, the construction, and fundamental properties. In particular, quantum natural time QLE has three phases analogous to those of SLE, and satisfies Markov properties identifying the random surfaces cut out by the process. Based on joint work with Deven Mithal.