You probably want to compare polygon "boundaries" with the linestring layer ?

I can see 2 tools for that :

1) Tools > QA > Calculate Geometry Differences...

2) Extension > Matching > Matching (needs to be installed)

First one is very efficient if you want to locate every small differences between shapes of two datasets supposed to be equal. Not sure it can answer you specific problem though (don't know if geometry are comapred with strict equality or topological equality).

The second one can find exactly what you are looking for (it has options to test geometric/topologic equality or 2D/3D equality), but you must first convert your polygons to linestring because linestring and polygons will never be considered equal.