In this talk, we address the high-dimensional noisy tensor estimation problem known as multi-spiked tensor PCA (Principal Component Analysis). This problem involves estimating a finite number of unknown, orthogonal signal vectors (referred to as spikes) from noisy tensor observations. I will present new results, obtained in collaboration with Gérard Ben Arous (NYU) and Cédric Gerbelot (NYU), on the sample complexity and running time required for stochastic gradient descent to efficiently recover all spikes from random initialization. Our results reveal a sequential recovery process, in which each spike is recovered one after the other, ultimately leading to either exact recovery of each spike or recovery of a permutation of the full set.