JK Flip Flop , Counter Sequence

The key thing is to treat each bit individually.

For example, for the low-order bit, lets call it next_state[0].

state     | next_state[0]
---------------
   000    | 0
   001    | 0
   010    | 1
   011    | X
   100    | 0
   101    | 0
   110    | 0
   111    | 1

So you can write

next_state[0] = state[0] & state[1] & state[2] | ~state[0] & state[1] & ~state[2]

If you are building this with discrete logic you could simplify this to

next_state[0] = state[1] & (state[0] & state[2] | ~state[0] & ~state[2])

And you have an equation you can use to drive the flip-flop that will generate the next condition for the low-bit of the state variable.

Finding the equations for the other two state bits works the same way.

Considering a positive-edge triggered FF followed by a negative-edge triggered FF and 1st FF is triggered at t=0 sec. The triggering edge for 2nd FF will arrive only after \$T_{clk} + T_{skew}\$. But the output of 1st FF takes some time (\$T_{clk\rightarrow Q}\$) to settle to a stable value. Stable value should reach at input to the 2nd FF atleast \$T_{setup}\$ seconds before the triggering edge arrives. ie,

$$\frac{T_{clk}}{2} + T_{skew} > T_{clk\rightarrow Q} + T_{logic,max} + T_{setup} $$ or, $$ T_{logic,max} < \frac{T_{clk}}{2} - T_{clk\rightarrow Q} - T_{setup} + T_{skew}$$

The triggering edge to 2nd FF arrives at \$t = T_{clk} + T_{skew}\$. The next triggering edge for 1st FF will happen at \$t =T_{clk}\$ sec, ie., \$T_{clk}/2-T_{skew}\$ sec after triggering 2nd FF. Hence 2nd FF can have a hold time of \$T_{clk}/2-T_{skew}+ min\ delay\ between\ the\ FF's\$. ie., $$T_{hold} < \frac{T_{clk}}{2} - T_{skew} + T_{clk\rightarrow Q,cd} + T_{logic,min}$$ or, $$ T_{logic,min} > \frac{T_{clk}}{2} + T_{clk\rightarrow Q,cd} + T_{hold} + T_{skew}$$

Where, \$T_{clk\rightarrow Q,cd}\$ is the clock to Q contamination delay defined as time taken to produce the first change in output of the FF after applying clock and \$T_{clk\rightarrow Q}\$ is the time taken to make stable change in the FF output after clock is applied.