PCP research proposal
Here are some recent papers on PCPs with small query complexity that I found interesting:


arxiv.org/pdf/1305.1979
eccc.hpi-web.de/report/2013/179/download
wisdom.weizmann.ac.il/~dinuri/mypapers/DH.pdf   
eccc.weizmann.ac.il/report/2015/085   
cs.utexas.edu/~danama/papers/par-rep/final3.pdf   
cs.utexas.edu/~danama/papers/par-rep-limit/paper.pdf   
eccc.hpi-web.de/report/2016/160


Also, this paper is somewhat older, but it is interesting and can help understanding the context to some of the previous papers. In addition, if I may recommend a paper of my own, I think this paper contains open problems that might be useful.