A big question in archeology and historical linguistics is if and how well the Neanderthals spoke. Our immediate ancestor species, Homo erectus, Homo heidelbergensis, Homo ergaster all are speculated to be have had language.
Anyhow, it occurred to me that an experiment could be done to decide what is the least sophisticated language necessary to accomplish the behaviors implied by the remaining material culture of erectus, et al. The ancient humans needed to speak well enough to teach the next generation how to flake stone, how to butcher animals, build a fire, build a tent out of mastadon bones with post holes, hunt in a pack with thrusting spears, sharpen spears, gather and use ocher for tanning or painting. If these can be done with no speech what-so-ever, just a theory of mind--i.e. I can see you are picking up your spear, you don't need to say, "mi mute o alasa e soweli moku!"--then the Neanderthals didn't need speech and the simplest theory is they didn't have it.
There are some challenges to such a test. It's also possible that all of these behaviors are like bird nest building. We can ask human volunteers to speak only in toki pona, but we can't tell them to stop using complicated semantic strategies, for example, "Did you enjoy "Annie Hall"?" -"All of Woody Allan's movies are great", requires some mental gymnastics to understand that the answer means (probably) yes. Worse, once the outlines of the experiment were explained to volunteers, they don't need to talk amongst themselves as much any more.
This sort of experiment isn't exactly a slam dunk proof of anything, after all, experiments with reed boats have demonstrated one can do incredible feats of sailing if you keep at it and know in advance that you're not on a suicide mission.
On the syntax and vocabulary side of things, we'd expect neanderthals (and other early humans) to have basic words for flake, to flake, scraper, core, hand-axe and point.