## 2 Theory and Computational Methods  
To calculate the intrinsic anomalous and spin Hall conductivities of a solid, we applied Kubo formalism. Following our previous studies [12], we can assert that the intrinsic Hall effect comes from the static limit (\(\omega=0\)) of the off-diagonal elements of the optical conductivity. We note that the anomalous Hall conductivity (AHC) calculated in this way is in good agreement with that calculated directly by accounting for the Berry phase correction to the group velocity. We first computed the imaginary part of the off-diagonal elements of the optical conductivity. Then, we obtained the real part of the off-diagonal elements from the corresponding imaginary part using Kramers-Kroning transformation. The intrinsic Hall conductivity \(\sigma_{xy}^{H}\) is the static limit of the off-diagonal element of the optical conductivity \(\sigma_{xy}^{(1)}\) (\(\omega=0\)) (see [12] for more details). The intrinsic Hall effects are driven by spin-orbit coupling (SOC), while the extrinsic Hall effects may be caused by the scattering of magnetic impurities, also known as skew-scattering or a side-jump contribution. For readers who are interested in this research field, [PERSON] et al. published a review article in Reviews of Modern Physics in 2010 [14]. Our first-principles calculations were based on fully relativistic band theory. The SOC effect was included in our theoretical calculations. We calculated the relativistic band structure of the Co\({}_{3-x}\)Cr\({}_{x}\)Al Heusler compounds using the projector augmented wave (PAW) method, as implemented in the Vienna ab initio simulation package (VASP) [15; 16]. The self-consistent electronic band structure calculations were based on density functional theory (DFT). Generalized gradient approximation (GGA) was applied to describe the exchange-correlation potential [17; 18]. We considered the fully ordered cubic Heusler compound structure (\(L2_{I}\)). The so-called regular and inversed (\(XA\)) Heusler structures differ by the arrangement of the X, Y and Z elements of the Heusler compounds. We will describe those differences later. The cut-off kinetic energy was set to 350 eV. To obtain an accurate ground state charge density, a very fine k-mesh grid of 30 \(\times\) 30 \(\times\) 30 was used. The self-consistent cycles were terminated when the total energy differences were less than 10\({}^{-6}\) eV. To obtain accurate AHC and SHC, a dense \(k\)-point mesh is required. We used several \(k\)-point meshes to extrapolate the AHC and SHC results in the infinite \(k\)-point mesh. To achieve this, we calculated the AHC and SHC as a function of the number of the \(k\) points (\(N_{k}\)) in the first Brillouin zone. The theoretical AHC and SHC results were fitted to a polynomial at \(N_{k}=\infty\). In our theoretical calculations, the charge densities were obtained using 30 \(\times\) 30 \(\times\) 30 k-mesh, which is far from sufficient to calculate the anomalous Hall and spin Hall conductivities. Alternatively, there is another method that can theoretically calculate the anomalous Hall conductivity, which is widely used by other research groups. The anomalous Hall conductivities are obtained by making use of the Berry connection. In principle, an infinite number of \(k\) points is also needed. This method typically makes use of Wannier interpolation [19], and, thus, a very high k mesh can be used. Our studiesalso relied on dense \(k\) points; however, it was very time-consuming and the disk space for recording momentum matrix elements increased rapidly. The finest \(k\) point used in the irreducible Brillouin zone was 267,228. We agree that these \(k\) points may still not be enough, and extrapolation from these calculated AHC and SHC may not be suitable. Our previous studies of the anomalous Hall conductivities in _fcc_ Ni [20], hcp Co [21] and Co-based Heusler alloys [12; 22] were in good agreement with other theoretical calculations and experiments, indicating that our approaches may be reliable.