Over the past two decades, ideas coming from quantum information science have become increasingly relevant in high-energy physics. This is particularly evident in an area of quantum gravity called holography, also known as the AdS/CFT correspondence, where two concepts which have led to novel insights are entanglement and complexity. Particularly compelling open problems in our understanding of these quantities include the time-dependence of complexity and the interplay between complexity and entanglement both in non-equilibrium systems and for mixed states in quantum field theories (QFTs). In this talk, we explore these problems in scenarios which allow us to make tractable computations and extract universal properties of these quantities. We first study the time-dependence of complexity in a free scalar theory undergoing a quench through a critical point, finding evidence for universal scaling behaviour. We then study generalizations of complexity and entanglement entropy to mixed states and study their universal properties, finding qualitative agreement with holographic expectations. Finally, we identify the long-distance behaviour of correlations in mixed states as an interesting limit and explores its properties, leading to a novel prediction across all conformal field theories with a gap in the operator spectrum. Collectively, these findings set the stage for a better understanding of complexity and entanglement in QFTs. This is paramount for elucidating the mechanism which connects gravity and quantum theories in holography. Consequently, we believe that our results can lead to a better understanding of quantum gravity and quite possibly to new tools in the study of quantum many-body systems.