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The complex structure of the human brain gives rise to emergent patterns of activity through the dynamic interactions among its various subcomponents. Recently, there has been growing interest in describing higher-order interactions beyond pairwise measures in brain data, particularly to stratify global conscious states. A promising approach is provided by Dynamical Independence (DI) - a formal information-theoretic measure quantifying the emergence of macroscopic dynamics. We applied this method to high-density EEG recordings in wakefulness, sleep and anaesthesia to quantify and characterise emergent dynamical structure. Applied to EEG, emergent dynamical structure is discovered by identifying coarse-grained variables as lower-dimensional spaces whose future evolution is not enhanced by knowing the past dynamics of their microscopic constituents. Our study reveals that emergent variables across all macroscopic scales in both propofol and xenon conditions show significantly higher degree of emergence than the wake condition. Conversely, ketamine shows significantly lower emergence than the wake condition across all macroscopic scales. Crucially we show that dynamical structure emerges in wake and ketamine-induced anaesthesia–where conscious report was recovered–across all high-order scales. Altogether, we present a new method to explore the emergent dynamical structure of global states of consciousness from EEG data, including sleep/wake transitions.