From the abstract:
How do protest actions impact public support for social movements? Here we test the claim that extreme protest actions—protest behaviors perceived to be harmful to others, highly disruptive, or both— typically reduce support for social movements. (…) participants indicated less support for social movements that used more extreme protest actions. (…) Negative reactions to extreme protest actions also led participants to support the movement’s central cause less, and these effects were largely independent of individuals’ prior ideology or views on the issue
Taken together with prior research showing that extreme protest actions can be effective for applying pressure to institutions these findings suggest an activist’s dilemma
The paper does not look at size of action, but method, such as use of violence, innocent victims, …
The paper suggests something else as more effective: small unknown groups benefit from violent action, as they’ve little to no support to lose. Later, when emotional response has died down, but having gained notoriety, they gain broader support with a more moderate messaging.
For well known causes, it just reduces long term support, but still makes it more likely for short term policy change.
EDIT: This is basically the origin story of most major religions, come to think of it