The paper investigates the dynamics of a free-fermion chain under dephasing, using two different random-measurement protocols: quantum-state-diffusion (QSD) and quantum-jump (QJ) unravelings. Both protocols preserve the state in a Slater determinant form, allowing for numerical analysis of systems with up to \( L = O(10^2) \) sites. The authors focus on the distribution of the expectations of the measured operators along quantum trajectories and observe a bifurcation in this distribution, where the shape changes from unimodal to bimodal. The bifurcation occurs at a similar measurement strength for both protocols, but the distributions and the transition properties differ due to the symmetries of the protocols. For both unravelings, the distribution shows a power-law scaling with the system size, indicating multifractal behavior. The study also examines the localization properties of the state, finding that the inverse participation ratio (IPR) scales with the system size as \( \sim L^{-\alpha} \) with \( 0 < \alpha < 1 \), consistent with multifractal behavior. The paper concludes by discussing the implications of these findings for the understanding of measurement-induced transitions and localization in monitored quantum systems.The paper investigates the dynamics of a free-fermion chain under dephasing, using two different random-measurement protocols: quantum-state-diffusion (QSD) and quantum-jump (QJ) unravelings. Both protocols preserve the state in a Slater determinant form, allowing for numerical analysis of systems with up to \( L = O(10^2) \) sites. The authors focus on the distribution of the expectations of the measured operators along quantum trajectories and observe a bifurcation in this distribution, where the shape changes from unimodal to bimodal. The bifurcation occurs at a similar measurement strength for both protocols, but the distributions and the transition properties differ due to the symmetries of the protocols. For both unravelings, the distribution shows a power-law scaling with the system size, indicating multifractal behavior. The study also examines the localization properties of the state, finding that the inverse participation ratio (IPR) scales with the system size as \( \sim L^{-\alpha} \) with \( 0 < \alpha < 1 \), consistent with multifractal behavior. The paper concludes by discussing the implications of these findings for the understanding of measurement-induced transitions and localization in monitored quantum systems.