Temporal Quantum Correlations and Leggett-Garg Inequalities in Multilevel Systems

We show that the quantum bound for temporal correlations in a Leggett-Garg test, analogous to the Tsirelson bound for spatial correlations in a Bell test, strongly depends on the number of levels N that can be accessed by the measurement apparatus via projective measurements. We provide exact bounds for small N that exceed the known bound for the Leggett-Garg inequality, and we show that in the limit N -> infinity the Leggett-Garg inequality can be violated up to its algebraic maximum.