The human brain operates in large-scale functional networks, collectively subsumed as the functional connectome1-13. Recent work has begun to unravel the organization of the connectome, including the temporal dynamics of brain states14-20, the trade-off between segregation and integration9,15,21-23, and a functional hierarchy from lower-order unimodal to higher-order transmodal processing systems24-27. However, it remains unknown how these network properties are embedded in the brain and if they emerge from a common neural foundation. Here we apply time-resolved estimation of brain signal complexity to uncover a unifying principle of brain organization, linking the connectome to neural variability6,28-31. Using functional magnetic resonance imaging (fMRI), we show that neural activity is marked by spontaneous "complexity drops" that reflect episodes of increased pattern regularity in the brain, and that functional connections among brain regions are an expression of their simultaneous engagement in such episodes. Moreover, these complexity drops ubiquitously propagate along cortical hierarchies, suggesting that the brain intrinsically reiterates its own functional architecture. Globally, neural activity clusters into temporal complexity states that dynamically shape the coupling strength and configuration of the connectome, implementing a continuous re-negotiation between cost-efficient segregation and communication-enhancing integration9,15,21,23. Furthermore, complexity states resolve the recently discovered association between anatomical and functional network hierarchies comprehensively25-27,32. Finally, brain signal complexity is highly sensitive to age and reflects inter-individual differences in cognition and motor function. In sum, we identify a spatiotemporal complexity architecture of neural activity — a functional "complexome" that gives rise to the network organization of the human brain.