"great quantities" is correct and usable in written English.
You can use it when you want to refer to an abundance of something. For example, "Great quantities of rainwater had collected in the ditch."
"It's not eaten in great quantities," Ms. Kennedy said.
Great quantities of hard reading are involved, and endless thought.
New Year's Eve? Funny hats and great quantities of alcohol.
Production was on a vast scale, and great quantities survive.
France in particular produced great quantities of superior faience tableware.
It exports great quantities of agricultural produce and livestock.
They provide beer, and sell drugs, mostly cocaine and marijuana, in great quantities.
If eaten in great quantities before a feast, there would be no hangover.
At the time, great quantities of art reached the public through the funnel of the Salon.
That would mean human stem cells, which can be grown in great quantities.
Progesterone is produced in the second half of the cycle and in great quantities in pregnancy.
