The assumption that conserved quantities, also known as charges, commute underpins many basic physics derivations, like that of the thermal state's form and Onsager coefficients. Yet, operators' failure to commute plays a key role in quantum theory, e.g. underlying uncertainty relations. Lifting this assumption has spawned a growing subfield of quantum many-body physics [1]. How can one argue that charges' noncommutation caused a result? To isolate the effects of charges' noncommutation, we created analogous models that differ in whether their charges commute and discovered more entanglement in the noncommuting-charge model [2]. We further introduce noncommuting charges (an SU(2) symmetry) into monitored quantum circuits, circuits with unitary evolutions and mid-circuit projective measurements. Numerically, we find that the SU(2)-symmetric model has a critical phase (a phase with long-range entanglement) in place of the area-law phase typically found in these circuits [3]. [1] Majidy et al. "Noncommuting conserved charges in quantum thermodynamics and beyond." Nat Rev Phys (2023) [2] Majidy et al. "Non-Abelian symmetry can increase entanglement entropy." PRB (2023) [3] Majidy et al. "Critical phase and spin sharpening in SU(2)-symmetric monitored quantum circuits." PRB (2023)