Multi-Order Dynamic Causality

• Multi-order dynamic causality is a framework that models dynamic, higher-order causal relations evolving across time, context, and quantum operations.
• It employs methods like process matrix formalism, OCTE, and neural-based latent graphs to capture synergistic and indirect influences beyond pairwise effects.
• Applications span from quantum circuits and time series analysis to econometrics, while addressing challenges in computational scalability and hidden confounders.

Multi-order dynamic causality refers to frameworks and methods that formalize, detect, and utilize causal relations that are (i) structured over multiple orders (i.e., including higher-order, synergistic or indirect influences, not only direct/pairwise ones) and (ii) dynamically evolving—either in time, context, or as a function of quantum operations and information flows. This concept is foundational in the analysis of complex systems ranging from multivariate stochastic processes to quantum circuits, where causality may be indefinite, dynamic, synergistic, or encoded in the structure of dynamical processes themselves. Below, the main lines of development and the principal formulations of multi-order dynamic causality are presented.

1. Formal Structures: Causal Orders, Dynamical and Multi-Order Extensions

Classical causal structures are typically captured by directed acyclic graphs (DAGs), where edges denote direct cause-effect relations. Multi-order dynamic causality generalizes this by allowing:

• Dynamical causal order: The causal structure itself can change during the system's evolution or be influenced by upstream choices. In quantum information and multi-party settings, this can include indefinite or contextually established causal orders (Castro-Ruiz et al., 2017, Mothe et al., 10 Jul 2025).
• Higher-order and synergistic interactions: Causality is not just encoded as pairwise links but may manifest as irreducible k-way (k>2) influences or through multi-hop associative chains (Kořenek et al., 2024, Cao et al., 26 Aug 2025).
• Multi-horizon or dynamic timescales: In longitudinal/time series data, causality can unfold along multiple timescales, introducing horizon-dependent causal networks (Dettaa et al., 2024, Sanchez et al., 28 Oct 2025).

The process matrix formalism in quantum theory makes this explicit, defining admissible sets of operations in which the underlying causal order can be definite, indefinite, or even a coherent superposition and subject to transformation by supermaps (Castro-Ruiz et al., 2017).

2. Quantum and Classical Hierarchies of Dynamic Causal Order

Multi-order dynamic causality is precisely classified within a hierarchy as established for both classical and quantum systems (Mothe et al., 10 Jul 2025):

• Static Order (Convex Fixed Order, convFO): Mixtures of correlations/processes compatible with a single total order.
• Non-influenceable Order (NIO/NIO'): Causal orders whose implementation weights are independent of inputs/actions of past parties, yet may be dynamically established "on the fly".
• General Causal Correlations: Those admitting recursive or hidden-order decompositions, allowing for choices of order conditioned on prior actions.
• Indefinite/Dynamical Causality (Quantum): Using process matrices and quantum circuits, control of the order may be classical, quantum, or partially decoupled from parties' local actions, supporting both dynamical and indefinite forms (Mothe et al., 10 Jul 2025, Castro-Ruiz et al., 2017).

For four or more parties, there are causal correlations and processes that have genuinely dynamical, non-influenceable causal order, which cannot be simulated by convex mixtures of static orders nor by classical switches with causal influence by previous parties (Mothe et al., 10 Jul 2025).

3. Higher-Order Causality: Information-Theoretic and Hypernetwork Approaches

Beyond pairwise interactions, multi-order causality requires tools that can faithfully identify (i) irreducible synergistic influences, and (ii) the absence of lower-order explanatory structures. The optimally conditioned transfer entropy (OCTE) framework formalizes this using conditional mutual information (Kořenek et al., 2024):

• OCTE: For a candidate set of sources $\mathcal{X}_I$ and target $Y$, one computes $I(\mathcal{X}_I; Y | S)$ for all $S \subseteq (\mathcal{X} \setminus \mathcal{X}_I)$. If $I(\mathcal{X}_I; Y | S) > 0$ for all $S$, and every strict subset of $\mathcal{X}_I$ fails this condition, $\mathcal{X}_I$ is deemed a direct causal set of order $k=|\mathcal{X}_I|$.
• Causal Hypernetworks: The complete causal structure forms a directed hypergraph, with (possibly irreducible) higher-order edges indicating true multi-way (synergistic) causal influence.

A canonical example is the XOR logic gate: neither input alone has predictive power, but the pair does—a purely second-order causal effect.

4. Dynamical Causality in Time Series and Point Process Models

In systems modeled by time series or event sequences, dynamical causality manifests as time-resolved, multi-step, and possibly horizon-dependent influence patterns.

• Multi-horizon Granger Causality: High-dimensional sparse VARs and local projections enable inference of causal relations at varying horizons. The main insight is that direct (first-step) Granger non-causality does not imply absence of indirect or delayed effects. Robust multi-horizon tests and network visualizations reveal how causal impact propagates across variables and timescales (Dettaa et al., 2024).
• Diffusion-Ordered Temporal Structure (DOTS): Multiple valid causal orderings in temporal DAGs, induced by stochastic diffusion-based score-matching and Hessian-based leaf detection, are aggregated to reconstruct the transitive closure and thus robustly recover the true (possibly multi-order) temporal causal structure (Sanchez et al., 28 Oct 2025).
• MOCHA for Temporal Point Processes: Multi-order dynamic causality is captured by explicit modeling of multi-hop (multi-step) causal paths in directed, time-varying latent graphs, learned end-to-end via neural methods. Both direct and indirect influences (up to length L) are encoded, and sparsity and acyclicity are enforced. The framework reveals time-evolving patterns of direct, indirect, and higher-order causal dependencies in complex event sequences (Cao et al., 26 Aug 2025).

5. Degrees-of-Freedom Signatures and Hidden Causal Structures

A complementary approach exploits the concept of effective degrees of freedom (df) in dynamical systems (Telcs et al., 2024):

• df-causality: For two observed subsystems, the minimum number of past states required to render the future independent of further history defines their degrees of freedom. The configuration of joint and marginal dfs classifies the causal configuration: independence, unidirectional drive, two-way feedback, or common hidden driver.
• This method robustly detects hidden confounders: if the joint df is strictly between the maximum and sum of the marginals, a hidden common cause is inferred. Empirical validation confirms the signatures align with both classical causal inference and Granger causality tests.

6. Process Matrix and Supermap Formalism: The Hierarchy of Higher-Order Maps

The process matrix formalism provides a rigorous foundation for quantum multi-order dynamic causality (Castro-Ruiz et al., 2017):

• Process Matrix ($W$): Encodes the most general input-output coupling among parties' operations, subject to positivity, normalization, and subspace constraints that forbid closed time-like curves.
• Supermaps: Transformations mapping process matrices to process matrices. Complete positivity and subspace preservation are required. All continuous, reversible supermaps are local unitaries and preserve causal order; non-reversible or discontinuous supermaps (e.g., the quantum switch) allow transitions to indefinite or altered causal order.
• Infinite Hierarchy: Iterating the supermap construction yields a tower of higher-order maps (maps of maps, etc.), governed by projectors $P^{(n)}$ at each level. This infinite ladder accommodates arbitrarily composable transformations of dynamic causal structure.
• Such a hierarchy unifies the analysis of composable quantum networks, adaptive protocols, and potentially emergent spacetime structures under quantum and indefinite causal constraints.

7. Applications, Limitations, and Open Questions

Multi-order dynamic causality finds broad application across quantum information, neuroscience, econometrics, and the modeling of complex temporal and event-driven systems. State-of-the-art frameworks (e.g., DOTS, MOCHA) yield superior empirical performance in structure recovery and predictive tasks compared to single-order or static alternatives (Cao et al., 26 Aug 2025, Sanchez et al., 28 Oct 2025). However, several open challenges remain:

• Physical characterizations of supermaps and process dilations that maintain validity under all intermediate evolutions (Castro-Ruiz et al., 2017);
• Robust detection and interpretation of higher-order and synergistic links in nonlinear or partially observed systems (Kořenek et al., 2024);
• Computational and statistical scaling in the exhaustive search for higher-order conditional dependencies or for multi-horizon dynamical networks;
• Formal unification of quantum and classical multi-order causality, including the experimental identification of genuinely dynamical but non-influenceable orderings (Mothe et al., 10 Jul 2025).

The further development of composable, interpretable, and scalable approaches to multi-order dynamic causality is both a methodological frontier and of substantive importance for modeling and controlling complex, hierarchically coupled systems.