Q:

Can you QuickSilver Sash or Cleanse out of Poppy's Ultimate?

The question is summed up in the title. What happens exactly?

A:

You cannot completely cleanse or quicksilver sash out of poppy's ultimate for a very specific reason:
Poppy's ultimate is NOT wholy a debuff on your champion. There are two parts:
One is a buff applied to poppy granting her certain invulnerability to other enemy units. This cannot be touched.
The other is a debuff on your champion increasing the damage you take from poppy. This CAN be QSSed.

