The crossbow appears to have become a major Chinese military weapon during the Spring and Autumn Period, during the 6th century at the latest. Bronze trigger mechanisms have been found in Chu royal tombs; royal tombs have the advantage of often being datable from literary sources, as long as the occupant can be identified. These predate the earliest known Greek crossbows.

However, this doesn't mean that the crossbow was invented in China. The early history of the crossbow, including its invention, is obscure, due to the poor archaeological preservation of wood and bamboo. It is exceedingly unlikely that the earliest crossbows had sophisticated bronze trigger mechanisms. These sophisticated crossbows were almost certainly an evolutionary development from older crossbows with wooden (or other organic) trigger mechanism. The adoption of bronze was probably driven at least in part by a push towards higher draw weights, to make crossbows more effective in military use (at which point, we can note that the Chinese tradition of using crossbows with long power strokes goes back to the Spring and Autumn Period, with crossbow bolts from this time exceeding 60cm in length).

At some stage, simple crossbows used for hunting, including use as traps, spread across northern Eurasia. Their origin and spread is invisible in both archaeology and literature, so we don't even have good guesses as to where they were invented, beyond the not-very-useful "northern Eurasia".

There are possible crossbow parts from Neolithic China. These are bone and shell strips from, e.g., the Yangshao culture, and are similar to trigger mechanism parts used in tribal crossbows in modern times. The Chinese Neolithic is relatively late, so we're only taking about the 4th or 5th millennium BC.

For brief discussion of these possible crossbow parts, and also Spring and Autumn crossbows, see

• Yang Hong, Weapons in Ancient China, Science Press, New York/Beijing, 1992. Neolithic crossbows on pg 16, Spring and Autumn crossbows on pp 95-97.

Without running contrary to /u/wotan_weevil about the overall dating of the crossbow, a more recent examination of the Shandong graves has suggested that they date much later than the original excavators believed. Pages 16-19 of the thesis provide a summary of the new findings, based on comparison of pottery with other burials in the region, and page 23 has a table that provides a handy guide to how the dates should be shifted. In this case, Graves M3 (p88-90) and M52 (p92-93), both originally dated to the "Early Warring States" period, most likely belong to the "Middle Warring States" period (390-305BC).

This doesn't mean that crossbows didn't exist before this, and it's quite likely that they did. The Romance of Wu and Yue, dating to the first century AD, dates the introduction of a militarily effective crossbow to some time before the mid-6th century, while references to the crossbow appear in Sun Tzu's Art of War (generally dated a couple of decades either side of 500 BC, though most likely 480-500 BC). Factoring in the extenant crossbow bolts from the Spring and Autumn Period mentioned by /u/wotan_weevil, it's clear that crossbows came into widespread use at some point during the 6th century BC at the very latest, and there is very chance they saw some use before then.

As to the invention of the "crossbow" in the West, the closest parallel would be the gastraphetes, which was developed at least by 399 BC and possibly earlier. This is a poor comparison, as the crossbow was a light, portable individual weapon, while the gastraphetes was more likely designed for use in sieges and never saw much (if any) use in the field. It's not until the 1st or 2nd centuries AD that we have any evidence of a weapon comparable in both design and usage to the Chinese crossbow, several hundred years after we find definitive archaeological traces in China.

Stephen Selby's Chinese Archery has a useful summary of the early written evidence for the crossbow, including several translations.