Violation of Macrorealism: Theory & Applications

• Macrorealism is defined by the claim that macroscopic systems possess definite observable values at all times, a notion directly challenged by quantum violations of Leggett–Garg inequalities and NSIT conditions.
• Experimental protocols employing coarse-grained, noninvasive, and bias-adjusted measurements reveal robust MR violations even under realistic noise and unsharp measurement scenarios.
• Extensions to many-valued and continuous-variable systems confirm that MR violations persist across diverse experimental setups, providing crucial insights for quantum foundations and emerging technologies.

Macrorealism (MR) is the conjunction of two central assumptions for macroscopic systems: (i) realism per se, wherein the system always possesses definite values for relevant observables at all times, and (ii) noninvasive measurability, the possibility of ascertaining those values without disturbing the system's subsequent evolution. Quantum mechanics predicts systematic violations of macrorealism in various scenarios, challenging the classical worldview that definite properties govern macroscopic dynamics. Detection and quantification of these violations underlie fundamental experimental and theoretical programs, with broad implications for quantum foundations, measurement theory, and emerging quantum technologies.

1. Theoretical Foundations and Formal Criteria

Macrorealism is operationalized through necessary conditions, typically defined as follows:

• Leggett–Garg inequalities (LGIs): For sequential measurements of a dichotomic observable $Q(t) \in \{+1, -1\}$ at three times, the two-time correlators $C_{ij} = \langle Q(t_i) Q(t_j) \rangle$ satisfy the LGI $K_3 = C_{12} + C_{23} - C_{13} \leq 1$ under MR and noninvasive measurements. Quantum systems can yield $K_3 > 1$, violating this bound.
• Wigner's form of the LGI (WLGI): Bounds on combinations of joint outcome probabilities such as $W = P_{23}(+,+) - P_{12}(-,+) - P_{13}(+,+) \leq 0$ also witness violations.
• No-signaling in time (NSIT): The marginal statistics at a later time must not depend on whether an earlier measurement was performed, formalized as $$\Delta = P(Q_3=-1) - [P_{23}(+,-) + P_{23}(-,-)] = 0$$ for dichotomic observables.

Violation of any of these points to incompatibility with macrorealism (Mal et al., 2016, Cai et al., 19 May 2025, Maroney et al., 2014, Kofler et al., 2012).

2. Violation Mechanisms and Robustness

Quantum mechanics predicts persistent and robust violations of MR in a variety of systems:

• Multilevel Spin Systems: For spin-$j$ systems, the violations of LGI, WLGI, and NSIT saturate their algebraic maxima in the large-spin limit with sharp projective measurements. The algebraic maximum, for example, is $K_3 - 1 \to 2$ as $j \to \infty$ under suitable measurement choices. Crucially, violations remain robust for arbitrary unsharp positive-operator-valued measures (POVMs) characterized by a sharpness parameter $\eta \in (0,1]$—for any $\eta > 0$, the violation persists near the algebraic maximum (Mal et al., 2016).
• Coarse-Graining and Outcome Grouping: Violations can be observed even when measurement outcomes are coarse-grained by grouping multiple levels into two sets, except in the special case of a perfectly symmetric grouping in the LGI scenario. WLGI and NSIT violations remain universal even under extreme coarse-graining. Asymmetry in outcome groupings is shown to be critical for maintaining LGI violations (Mal et al., 2016).
• Measurement Parameter Dependencies: In generalized measurement models parameterized by sharpness ($\eta$) and bias ($b$), the LGI/WLGI/NSIT quantum violations are decreased by unsharpness but can be restored or enhanced by introducing bias in the measurement apparatus. NSIT-violation can, in some protocols, be achieved for arbitrarily unsharp yet biased measurements (Das et al., 2017, Das et al., 2017).

3. Extensions to Many-Valued and Continuous-Variable Systems

Violations of MR are not restricted to dichotomic observables. Recent work extends the discourse to:

• Many-Valued and High-Dimensional Systems: LGI and NSIT generalizations to systems with trichotomic (qutrit) or higher-level observables show a breakdown of the simple dichotomic LG–NSIT hierarchy. For instance, in multi-level spins or qutrits, partial NSIT can coexist with LGI violations, and degeneracy-breaking measurements can yield violations even beyond the Lüders/Tsirelson bound, sometimes necessitating the full set of many-valued NSIT conditions for a complete macrorealist diagnosis (Halliwell et al., 2020, Mawby, 26 Feb 2024).
• Continuous-Variable Systems: Quantum harmonic oscillators and massive objects (even with macroscopic mass) provide settings where coarse-grained position measurements (with accuracies much worse than the SQL) can demonstrate quantum MR-violation. Such violations can be made independent of system mass, frequency, or initial momentum, showing that quantum behavior can persist at arbitrary mass scales, including regimes relevant for LIGO-scale mirrors (Das et al., 2022, Mawby, 26 Feb 2024).

4. Experimental Protocols, Decoherence, and Noise

Experimental realization of MR-violation tests adapts to constraints of measurement invasivity, decoherence, and practical imperfections:

• Noninvasive and Loophole-Free Strategies: Negative-result (interaction-free) measurements, quantum non-demolition protocols, and exact weak-measurement models permit tests that minimize quantum backaction or operational "clumsiness" (Wang et al., 2023, Vitagliano, 2015, Cour, 2022, Melegari et al., 24 Feb 2025). Quantitative invasivity quantifiers, such as addition of a clumsiness term to the Leggett-Garg bound, provide operational tightness against classical loopholes (Vitagliano, 2015).
• Macroscopic Ensembles and Quantum Computing: Scale-invariant violations can be achieved in macroscopic qubit ensembles via sequential parity measurements. In the ideal, decoherence-free limit, the amount of violation is independent of the ensemble size $N$, but realistic noise induces a quantum-to-classical transition as $N$ increases, providing a quantumness benchmark for large quantum processors (Braccini et al., 19 Nov 2025, Zindorf et al., 19 Nov 2025). Noise suppression, precise rotation control, and coupling homogeneity are shown to be critical for maintaining observable violations up to $N \sim O(10^2)$ (Braccini et al., 19 Nov 2025).
• Classical Models and Macrorealism: Classical hidden-variable models armed with contextual post-selection can, in specific settings, reproduce LGI violations while remaining locally realist, making the distinction between MR and local realism subtle. Violation of LGI or WLGI in such cases rules out MR+NIM, not local realism per se (Cour, 2022).

5. Beyond Standard LGI: Entropic and High-Order Approaches

The landscape of macrorealism violation extends beyond the canonical LGI:

• High-Order and Entropic Leggett-Garg Inequalities: Incorporation of higher-order temporal correlations and geometric methods yields entropic LGIs (ELGIs), which probe temporal orderings and conditional mutual informations. In large-dimensional (e.g., large-spin) systems, ELGI violations typically approach a constant with respect to the system's entropy, effectively rendering them negligible in the macroscopic limit—unless particular temporal scheduling brings about singular regimes with maximal quantum violations, even for maximally mixed states (Cai et al., 19 May 2025). This quantifies the quantum-to-classical transition in temporal correlations.
• Single-Measurement and Shared Randomness Protocols: It is possible, under shared randomness assumptions, to construct single-measurement witnesses of macrorealism violation, exposing classes of "macroscopic no-signaling" models via tailored inequalities (Sarkar, 2023).

6. Interpretational Issues and Partial Macrorealism

Violation of LGI and related inequalities does not always amount to ruling out all species of macrorealist positions. Several subtle distinctions arise:

• Operational Non-Disturbance vs. Ontic Noninvasiveness: LGI violations eliminate only the narrowest macrorealist class—those imposing operational eigenstate mixture MR—while more inclusive forms (eigenstate-support MR, supra-eigenstate-support MR) are untouched unless additional constraints (such as ontic noninvasiveness guaranteed by detailed device modeling) are justified (Maroney et al., 2014).
• Weak Macroscopic Realism (wMR): Quantum systems can violate LGI while still permitting a "weakened" macrorealism in which outcome predetermination is only ascribed after the measurement basis has been set via a reversible unitary, but not for all possible settings simultaneously. Leggett-Garg–Bell violations may remain compatible with such wMR, even while deterministic MR is empirically excluded (Fulton et al., 2021).

7. Implications, Outlook, and Physical Significance

The persistence and robustness of quantum violations of macrorealism in high-dimensional, noisy, or coarse-grained systems carry several key implications:

• Quantum-Classical Boundary: The absence of emergent classicality in the macroscopic limit (without explicit invocation of decoherence or time-coarsening) highlights that the breakdown of MR is not rescued by simple system size, measurement fuzziness, or coarse-graining alone (Mal et al., 2016, Cai et al., 19 May 2025).
• Metrological and Technological Relevance: Macroscopic tests of MR act as stringent diagnostics for quantumness in large-scale quantum devices and platforms, offering avenues for benchmarking and certifying truly nonclassical system dynamics (Zindorf et al., 19 Nov 2025, Braccini et al., 19 Nov 2025).
• Foundational Insights: The framework refines the conceptual distinction between temporal and spatial nonclassicality, the respective prerequisites for violating macrorealist or local realist inequalities, and their differing operational and ontological underpinnings (Maroney et al., 2014).
• Future Directions: Further exploration includes the deployment of necessary and sufficient quasi-probability–based tests; development of measurement protocols for high-dimensional, continuous, and many-valued systems; and empirical efforts in pushing MR-violation signatures into ever more macroscopic and technologically challenging domains (Melegari et al., 24 Feb 2025, Das et al., 2022, Roldán et al., 2017).

In summary, the quantum mechanical violation of macrorealism remains robust and theoretically unavoidable across a wide vista of system sizes, measurement models, and experimental scenarios, fundamentally challenging the classical notion that macroscopic objects always possess definite properties unaffected by measurement (Mal et al., 2016).