Despite the Sun standing out as the brightest source in the (sub)-millimeter sky, observing it presents significant challenges due to the rapid evolution of its complex small-scale structure on very short timescales. However, solar observations at (sub)-millimeter wavelengths offer immense diagnostic potential, as the continuum radiation in this range is closely linked to the electron temperature of the emitting layer. Additionally, with decreasing frequency, the (sub)-millimeter continuum originates from increasingly higher layers in the Sun's atmosphere. Fully exploiting these properties would enable the tracking of propagating phenomena such as waves and even perform three-dimensional tomography of the thermal structure with high temporal resolution. The anticipated results would have profound implications for understanding the fundamental physical processes behind the heating and activity of the solar atmosphere, and consequently, for comprehending stellar activity and its effects on exoplanet habitability. Unlocking this scientific potential necessitates expanded instantaneous frequency coverage beyond ALMA's current capabilities. By employing numerical simulations, the presentation explores the potential of solar observations with increased instantaneous spectral bandwidth and compares this approach to simultaneous observations with sub-arrays.