We formulate a spatial extension of the Leggett-Garg inequality by considering three distant observers locally measuring a many-body system at three subsequent times. The spatial form is especially suitable to test the ability of quantum devices to generate spatiotemporal correlations in a Hamiltonian-agnostic manner. We illustrate our proposal for a Heisenberg chain in a magnetic field, showing indeed that the first inequality-violation time scales proportionally to the distance between measuring parties. We attribute this phenomenon to Lieb-Robinson physics and, confirming this connection, we find that violations are anticipated when increasing the interaction range. The inequality violation is readily observable in current quantum simulation platforms, particularly Rydberg atoms' arrays and ultracold atoms in optical lattices. In outlook, spatial Leggett-Garg inequalities constitute a practical tool for benchmarking the complex nonrelativistic dynamics of many-body quantum systems.

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• Quantum correlations in quantum information
• Quantum correlations, foundations & formalism
• Quantum entanglement
• Quantum information theory
• Quantum simulation
• 1-dimensional spin chains
• Trapped atoms
• Quantum spin chains
• Schroedinger equation
• Spin chains