On live server the cart is working correctly. I change the quantity of a product in the cart then I push the "Update cart" button and all works fine. I transferred the site another domain and the cart quantity doesn't work anymore. As I change the quantity of a product and click the button the quantity remains the same. If I go back and try to add the same product to the cart, even if I specify a different quantity like 10 and not 1, in the cart only 1 item is added every time I click on "add to the cart" button and not 10 as specified. I cannot figure out where's the problem. I excluded my custom theme and rolled back to the default theme and the problem persist. this site other functionality work fine.
The below things you could do to make it work:
Clear all caches from System->Cache Management & try in Private window.
If you are using any third party module for related with cart update Add to cart etc. Disable that.
Check your log files in the directory var/log & see if any known error is there.
At last go to your cart controller & check the add & update action in there.
In your Theme
place on line after getUrl('checkout/cart/updatePost') ?>" method="post"> paste the below code
<?php echo $this->getBlockHtml('formkey'); ?>