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We investigate violations of Leggett-Garg inequalities (LGIs) for a harmonicoscillator and a (1+1)-dimensional chiral scalar field with coherent-stateprojectors, which is equivalent to a heterodyne-type measurement scheme. Forthe harmonic oscillator, we found that the vacuum and thermal states violatedthe LGIs by evaluating the two-time quasi-probability distribution function. Inparticular, we demonstrate that the value of the two-time quasi-probabilityreaches -0.123 for a squeezed coherent-state projector, which is equivalent to98% of the L\"uders bound corresponding to the maximal violation of the LGIs.We also find a violation of the LGIs for the local mode of a quantum chiralscalar field by constructing a coherent-state projector similar to the harmonicoscillator case. In contrast to the harmonic oscillator, the periodicity in thetime direction of the quasi-probability disappears, which is related to theexistence of quantum entanglement between the local mode and its complementarydegrees of freedom.

Large violation of Leggett-Garg inequalities with coherent-state projectors for a harmonic oscillator and chiral scalar field