Fractal Geometry of Quantum Correlation Space: Scale-Dependent Dimensionality, Long-Range Dependence, and Signatures in Time Delay Fluctuations

We develop a metric-measure geometric reinterpretation of quantum entanglement in which nonlocal correlations are encoded in the intrinsic geometry of composite state space rather than in dynamical propagation. Using the Bures metric and CPTP coarse-graining semigroups, we define a scale-dependent effective dimension of the entangled-state manifold. We propose that geometric observables along the renormalization flow may exhibit long-range dependence (LRD) characterized by a Hurst exponent H ∈ (1/2,1). A conjectured infrared attractor H* ≈ 0.65 is formulated as a falsifiable hypothesis. The framework preserves no-signaling and does not modify quantum mechanics. A direct link to the variance (RMS) of Eisenbud–Wigner–Smith time delay is proposed, providing operational accessibility of geometric LRD effects.