Unpowered 74ls chips vs Powered Nano?
I would not worry about damaging the inputs by applying voltage from your Nano when the LS chips are not powered.  It would be good practice to configure the Nano's outputs to inputs whenever you power down the LS circuitry to save power.  To learn about the input and output circuitry of LS logic, you can read about it at http://www.ti.com/lit/an/sdya009c/sdya009c.pdf and http://ecee.colorado.edu/~mcclurel/ON_Semiconductor_LSTTL_Data_DL121-D.pdf .  Good luck!