One of the $\textit{Euclid}$ mission's key projects is the so-called 3$\times$2pt analysis, that is, the combination of cosmic shear, photometric galaxy clustering, and galaxy-galaxy lensing. Although $\textit{Euclid}$ has established quality requirements for the photo-$z$ accuracy needed for the weak lensing galaxy sample, no such requirements have been set for the photometric clustering sample. In this paper, we investigate the impact of redshift uncertainties on $\textit{Euclid}$'s photometric galaxy clustering analysis and its combination with weak gravitational lensing, focusing on data release 1 (DR1). In particular, we study whether having precise knowledge of the mean of the redshift distributions per bin is sufficient to avoid biases in the resulting cosmological constraints or whether accuracy in the higher-order moments of the distribution is required. We evaluate the results based on their constraining power on $w_{\mathrm{0}}$ and $w_{a}$ and define thresholds for the precision and accuracy of $\textit{Euclid}$'s redshift distribution of the photometric clustering sample. We find that the redshift distributions of the photometric clustering sample must be known at an accuracy of 0.004(1+$z$) in the mean in order to recover 80$\%$ of the constraining power in $\textit{Euclid}$'s DR1 $w_{\mathrm{0}}w_{a}$CDM 3$\times$2pt analysis. The impact of the uncertainty on the width is negligible, provided the mean redshift is constrained with sufficient accuracy. For most sources of redshift distribution error, attaining the requirement on the mean will also reduce uncertainty in the width well below the required level.
We present a distance-resolved reconstruction of the local line-of-sight Galactic magnetic field, $B_{||}$, by combining a 3D electron density map, $n_e$, derived from dust-map-informed simulations, with a full-sky map of Faraday rotation measure (RM). Our forward model evaluates RM on the same 3D grid as the $n_e$ map and compares it to the Galactic Faraday rotation sky. We infer $B_{||}$ using a Gaussian-process prior whose power spectrum is inferred from the data with geometric variational inference. The result is a local map of $B_{||}$, with uncertainties, within 1.25 kpc for $b>5$ deg. The reconstructed RM sky reproduces prominent features of the Faraday rotation sky, with a root-mean-square average field strength of $1.63 \pm 0.16$ $\mu$G. In face-on views, the magnetic field shows coherent patches with alternating sign and hints of kpc-scale modulations, together with significant structure on scales of order 100 pc. The reconstructed $B_{||}$ field has a 3D power spectrum with an average slope of $-2.73 \pm 0.19$. We validate the $B_{||}$ reconstruction using Galactic pulsars. Predicted RMs, computed by integrating $n_e B_{||}$ to each pulsar distance, correlate with observed RMs. Predicted dispersion measures from the $n_e$ map also correlate with measured DMs, although with significant scatter.