what is nyquist rate of $h(t)\cdot h(t)$ and $h(t)\circledast h(t)$ Let's say we have $h_c(t)$ as a continuous-time signal with bandwidth $B$ and we would like to sample it.
To be able to reconstruct it correctly, the sampling rate must be greater than $2B$.
Now assume we have $h_{c1}(t)=h_c(t)\cdot h_c(t)$ and $h_{c2}(t)=h_c(t)\circledast h_c(t)$ what is Nyquist rate of these two signals? Is it $4B$ for both of them because each $h_{c}(t)$ needs $2B$?
squaring a signal effectively doubles the bandwidth of the signal (and doubles the Nyquist rate, which is the lower limit of the sampling rate).  convolving a signal with anything, even itself, does not in general increase the bandwidth.