Spacetime as Emergent Geometry from Quantum Genealogy: The Quantum Family Tree Theory

Abstract We propose a framework in which spacetime geometry emerges from the genealogical structure of quantum entanglement. Beginning from a single entangled state that undergoes recursive binary splitting, mutual information between any two descendant particles decays exponentially with their genealogical distance ng (total hops through the tree via the lowest common ancestor). Defining k = ng/2 as generations from the common ancestor, the classical kinship prediction is R(k) = (1/2)k−1; our simulations reveal steeper quantum decay with decay exponent α that converges to 2.0 across depths 2 through 10. Physical distance emerges as the logarithm of inverse entanglement: d = a ln(1/E) + b. We present computational results from quantum simulations at depths 2 through 10 (up to 1,024 leaves, 20-tree ensembles) demonstrating: (1) monotonic kinship decay of mutual information across all genealogical distances at all depths, (2) a logarithmic distance-entanglement relationship with R2 = 0.993, consistent with logarithmic (RT-like) holographic-style scaling in this discrete model, and (3) structural universality across random quantum dynamics. The model aligns with and extends Van Raamsdonk's entanglement-spacetime correspondence, providing a concrete genealogical mechanism for emergent geometry.