  Holographic entanglement entropy was recently recast in terms of Riemannian
flows or 'bit threads'. We consider the Lorentzian analog to reformulate the
'complexity=volume' conjecture using Lorentzian flows -- timelike vector fields
whose minimum flux through a boundary subregion is equal to the volume of the
homologous maximal bulk Cauchy slice. By the nesting of Lorentzian flows,
holographic complexity is shown to obey a number of properties. Particularly,
the rate of complexity is bounded below by conditional complexity, describing a
multi-step optimization with intermediate and final target states. We provide
multiple explicit geometric realizations of Lorentzian flows in AdS
backgrounds, including their time-dependence and behavior near the singularity
in a black hole interior. Conceptually, discretized flows are interpreted as
Lorentzian threads or 'gatelines'. Upon selecting a reference state, complexity
thence counts the minimum number of gatelines needed to prepare a target state
described by a tensor network discretizing the maximal volume slice, matching
its quantum information theoretic definition. We point out that suboptimal
tensor networks are important to fully characterize the state, leading us to
propose a refined notion of complexity as an ensemble average. The bulk
symplectic potential provides a specific 'canonical' thread configuration
characterizing perturbations around arbitrary CFT states. Consistency of this
solution requires the bulk satisfy the linearized Einstein's equations, which
are shown to be equivalent to the holographic first law of complexity, thereby
advocating for a principle of 'spacetime complexity'. Lastly, we argue
Lorentzian threads provide a notion of emergent time. This article is an
expanded and detailed version of [arXiv:2105.12735], including several new
results.
