Pulling together the discussions from the comments: Overall? There's a bit of truth in both, but in both cases the statements are technically false and also ignore important context.
For the 600 beheaded Jews:
According to Mohammed al-Ghazali, during that time the Muslims allowed
the Jews who had refused to betray the Prophet during the Battle of
the Ditch to leave and "go wherever they wished". Sa'd ibn Mu'adh, a
companion of Muhammad, was chosen by him as an arbiter and all parties
agreed to abide by his judgment. Sa'd gave his verdict that "the men
should be killed, the property divided, and the women and children
taken as captives". Muhammad approved of the ruling, calling it
similar to God's judgment, after which nearly all male members of the
tribe who had reached puberty were beheaded The Muslim jurist Tabari
quotes 600–900 being executed. The Sunni hadith do not give the number
killed, but state that all pubescent males were killed and one woman.
This quote from the very heavily footnoted wikipedia article at https://en.wikipedia.org/wiki/Invasion_of_Banu_Qurayza (thanks to @Henry) The sources drawn from are all from various Muslim theologians and historical figures, and should be taken as such. Given best information from those sources, then, we have 600 or more Jews being beheaded. Muhammed did not do the beheading himself, and was not even the one making the decision that they be beheaded, but he did approve of the ruling that they be beheaded. It is at least plausible that he would have made the same ruling himself, if he were the one called on to do so. The sources available also present those Jews as having been at war with Muhammed's people at the time (almost certainly true) and as having betrayed them previously. (That part is possible, but by no means certain. This is the sort of thing that biased historians and theologians have been known to change.)
Additionally, beyond all uncertainty of textual source, beheading 600 Jews by himself would have required one beheading a minute for 10 hours - something that would seem unlikely for reasons of physical stamina alone.
For the torturing and killing unbelievers, I will refer to the quote presented by @DavePhD's answer. Again, Muhammed did not do the torturing or killing himself, though in this case it was at his command. Additionally, it was apparently done as punishment for acts of murder and severe violation of hospitality. According to the theological consensus of Muslim scholars, they were unbelievers only because the act of the murder inherently rendered them as such. Again, we are working from religious texts here, but it is highly unlikely that the original claim had anything more definitive to work off of, given its source.
In the broader sense, then, it is reasonable to conclude that he was at least partly responsible for the beheading deaths of hundreds of Jews, and that he was responsible for at least a few cases of torturous execution (some or all of which occurred to those who were no longer considered believers), but that the precise claims made by the billboard are not supported by the available evidence.