  The power of quantum sensing rests on its ultimate precision limit,
quantified by the quantum Cramer-Rao bound (QCRB), which can surpass classical
bounds. In multi-parameter estimation, the QCRB is not always saturated as the
quantum nature of associated observables may lead to their incompatibility.
Here we explore the precision limits of multi-parameter estimation through the
lens of quantum geometry, enabling us to experimentally evaluate the QCRB via
quantum geometry measurements. Focusing on two- and three-parameter estimation,
we elucidate how fundamental quantum uncertainty principles prevent the
saturation of the bound. By linking a metric of "quantumness" to the system
geometric properties, we investigate and experimentally extract the attainable
QCRB for three-parameter estimations.
