Quantum geometry, a quantum mechanical quantity comprised of Berry curvature and quantum metric, describes the geometric structure of the electronic bands in solids. The correlation between nontrivial quantum geometry and quantum materials leads to new findings in condensed matter systems. Here we demonstrate that altermagnets, with spontaneously broken time-reversal \(({\mathcal{T}})\) –half-lattice-translation and parity-time symmetry, host both \({\mathcal{T}}\) -odd and \({\mathcal{T}}\) -even quantum geometric quantities that simultaneously manifest themselves despite the vanishing net magnetization. Consequently, giant room-temperature third-order electrical transport responses with sizable quantum geometric contributions are observed in (101)-oriented RuO2 thin films, an altermagnetic candidate; in particular, the third-order Hall effect is intimately correlated with altermagnetic order and can serve as a promising tool for detecting the Néel vector. Our work not only supports the existence of altermagnetism in 8-nm-thick RuO2 thin films, but also shows altermagnets as a versatile platform for exploring quantum geometry and constructing quantum electronic and spintronic devices.
