This is what wikipedia has to say about the "quaestor"
The position served different functions depending on the period. In the Roman Kingdom, quaestores parricidii (quaestors with judicial powers) were appointed by the king to investigate and handle murders. In the Roman Republic, quaestors (also pluralized quaestores) were elected officials that supervised the state treasury and conducted audits.
Did the role of investigating murders go to any other particular magistrate or other person during the Republic? The media I've seen about this time period makes it seem like it was just constant, lawless revenge. How accurate is that idea?
Googling for this brought up this past post:
And other posts I've found on other sites make it sound like there wasn't much formal investigating or prosecution, but the posts also tend to focus on the Empire.
In the Republic, if someone was murdered, and was notable enough for anyone to care about it, in urban Rome, would it be the military who investigated it, or was there a civil magistrate that might be in charge? Or nobody?
Thank you!
Civil magistrates were in charge but not in the way you're thinking.
The person dying was arguably the most important element. It would be the job of the civil magistrates to make an argument about the significance of the death, and once people thought it was significant, it would be a community effort to resolve the crime.