How we can show that 2 correlated videos are similar? Can I tell that different versions of the same video are correlated to each other? 
and if so, how can I show this correlation? for example,
high-quality and poor-quality versions of the same video.
Can we say various episodes of the same tv series are correlated since they mostly share the same background, characters, and scene?
and if so, how I can extract this similarity?
Maybe it's better to  ask, how can I compare two videos and show that they are statistically similar or they are correlated?
The video is actually a sequence of images (frames, for a PAL standard that is 25 frames) and in such a sequence you can apply objective measures of quality to the certain number of frames. The most frequently used quality measures are PSNR (Peak Signal to Noise Ratio)1  and SSIM (Structural Similarity Index) 2.
In the matlab, there is a built-in fuction that calculate PSNR and SSIM:

psnr(image, ref_image) PSNR in Matlab
ssim(image, ref_image) SSIM in Matlab