  This Thesis explores the notion of Krylov complexity as a probe of quantum
chaos and as a candidate for holographic complexity. The first Part is devoted
to presenting the fundamental notions required to conduct research in this
area. Namely, an extensive introduction to the Lanczos algorithm, its
properties and associated algebraic structures, as well as technical details
related to its practical implementation, is given. Subsequently, an overview of
the seminal references and the main debates regarding Krylov complexity and its
relation to chaos and holography is provided. The text throughout this first
Part combines review material with original analyses which either intend to
contextualize, compare and criticize results in the literature, or are the
fruit of the investigations leading to the publications on which this Thesis is
based. These research projects are the subject of the second Part of the
manuscript. In them, methods for the efficient implementation of the Lanczos
algorithm in finite many-body systems were developed, allowing to compute
numerically the Krylov complexity of models like SYK or the XXZ spin chain up
to time scales exponentially large in system size. It was observed that the
operator Krylov complexity profile in SYK, a paradigmatic low-dimensional
chaotic system with a holographic dual, agrees with holographic expectations,
while in the case of integrable models like XXZ complexity is affected by a
novel localization effect in the so-called Krylov space which hinders its
growth. Finally, an exact, analytical, correspondence between the Krylov
complexity of the infinite-temperature thermofield double state in the
low-energy regime of the double-scaled SYK model and bulk length in the theory
of JT gravity is established.
