It's interesting that states formed in six different regions around the world at roughly the same time (3000 - 0 BCE) https://en.wikipedia.org/wiki/State_formation.
From what I understand, it wasn't until the end of the last glacial period that changes in the climate enabled humans to develop agriculture and settle into permanent communities.
It looks like this led to steady population growth, with states developing independently in different regions of the world as population density increased to a point able to sustain complex societies.
During the glacial period, most water was held up in ice caps, so more land was either desert or tundra than currently. On the other hand, it's not the case that everywhere was inhospitable for agriculture -- only that favourable regions were less common than today, and in different areas. That makes it seem strange that humanity collectively waited until the same time period in different areas of the globe to settle down.
Is it possible that complex societies developed during this period, possibly in areas which are now underwater? I'm not talking "Ancient Atlantis" pseudoscientific theories, but perhaps societies on the level of North American chiefdoms, with multiple settlements and agriculture.
This is a frequently asked question, and we had a very similar thread recently. Take a look at the answers by /u/Brigantus and myself.
By way of summary, it is possible that some society like that could have existed, but given everything we know currently is highly improbable.
Given that we have significant amounts of archaeological evidence for foraging societies, especially in the upper Paleolithic, it would be surprising for us to lack any evidence a society that should have a greater archaeological footprint (a chiefdom, like you postulate). That we should have so much evidence for societies with smaller archaeological footprints, yet none for societies with theoretically much larger footprints is pretty damning evidence from an archaeological stand-point.
The problem is that in archaeology we never seek to disprove anything - given the incomplete and partial nature of the archaeological record, it would be difficult to conclusively disprove any assertion. This is why I say that the existence of such a complex, Pleistocene society is possible. However, this doesn't mean anything goes in archaeology: we can use the evidence available to us to describe the most probable explanation and the likelihood that is the correct explanation. This is why I say is is highly improbable that such a Pleistocene society could exist, given all the evidence we have available to us. Unfortunately, this lack of "certainty" in a more scientific sense leaves the doors open for the kinds of pseudoarchaeology you mention, where they claim that we can't disprove that aliens built the pyramids, for instance. While they are correct in saying we cannot disprove that, that doesn't mean we can't say that is a highly unlikely explanation for the evidence we have.