Long-lived exciton coherences have been recently observed in photosynthetic complexes via ultrafast spectroscopy, opening exciting possibilities for the study and design of coherent exciton transport. Yet, ambiguity in the spectroscopic signals has led to arguments against interpreting them in terms of exciton dynamics, demanding more stringent tests. We propose a strategy, quantum process tomography (QPT) for ultrafast spectroscopy, and apply it to reconstruct the evolving quantum state of excitons in double-walled supramolecular light-harvesting nanotubes at room temperature from eight narrowband transient grating experiments. Our analysis reveals absence of nonsecular processes, unidirectional energy transfer from the outer to the inner wall exciton states, and coherence between those states lasting about 150 femtoseconds, indicating weak electronic coupling between the walls. Our work constitutes the first experimental QPT in a "warm" and complex system and provides an elegant scheme to maximize information from ultrafast spectroscopy experiments.