  Holonomic phases---geometric and topological---have long been an intriguing
aspect of physics. They are ubiquitous, ranging from observations in particle
physics to applications in fault tolerant quantum computing. However, their
exploration in particles sharing genuine quantum correlations lack in
observations. Here we experimentally demonstrate the holonomic phase of two
entangled-photons evolving locally, which nevertheless gives rise to an
entanglement-dependent phase. We observe its transition from geometric to
topological as the entanglement between the particles is tuned from zero to
maximal, and find this phase to behave more resilient to evolution changes with
increasing entanglement. Furthermore, we theoretically show that holonomic
phases can directly quantify the amount of quantum correlations between the two
particles. Our results open up a new avenue for observations of holonomic
phenomena in multi-particle entangled quantum systems.
