Answer: I'm not sure.
Benzalkonium chloride and benzethonium chloride do an excellent job killing bacteria. However, the research that I have found, shows that they are NOT very good at killing norovirus surrogates. Scientists have had difficulty growing the real human norovirus in a petri dish, so researchers have to use closely related virus "surrogates" to do the testing (murine norovirus and feline calicivirus). Then they HOPE the results would be the same for human norovirus. This research paper, (http://www.ncbi.nlm.nih.gov/pubmed/22897744) shows that benzalkonium chloride kills the norovirus surrogates (feline calicivirus and murine norovirus) but only at a relatively high concentration (.5%) and it took 2 hours! That would not be an effective hand sanitizer or very useful cleaning product because it takes too long. However, the My-Shield Hand Sanitizer which has test data which shows that it kills murine norovirus has the active ingredient of .13% Benzalkonium chloride. However, other ingredients in the hand sanitizer could be making it more effective. I don't know for sure. In this paper (http://www.ncbi.nlm.nih.gov/pubmed/22293670), researchers tested benzethonium chloride against murine norovirus and it did not kill it. No one really knows if the real human norovirus would behave the same. However, benzethonium chloride is one of the ingredients in Zylast antiseptic hand sanitizer which does claim to kill norovirus. I think perhaps the combination of the alcohol and the benzethonium chloride make it so effective. I am not sure. To learn more about Zylast, read this page about hand sanitizers that kill norovirus. However, there has been a recent breakthrough and scientists are figuring out how to grow the real human norovirus in the lab. I hope that soon hand sanitizers and cleaning products can be tested against the real human norovirus. I don't think we will know the real answers until products can be tested on the real human norovirus. Until then we have to make our best educated guesses using the data we have about the norovirus surrogates.