We use $N$ -body simulations to study halo assembly bias (i.e., the dependence
of halo clustering on properties beyond total mass) in the density and
primordial non-Gaussianity (PNG) linear bias parameters ${b}_{1}$ and ${b}_{\varphi}$ ,
respectively. We consider concentration, spin and sphericity as secondary halo
properties, for which we find a clear detection of assembly bias for ${b}_{1}$ and
${b}_{\varphi}$ . At fixed total mass, halo spin and sphericity impact ${b}_{1}$ and
${b}_{\varphi}$ in a similar manner, roughly preserving the shape of the linear
${b}_{\varphi}({b}_{1})$ relation satisfied by the global halo population. Halo
concentration, however, drives ${b}_{1}$ and ${b}_{\varphi}$ in opposite directions. This
induces significant changes to the ${b}_{\varphi}({b}_{1})$ relation, with higher
concentration halos having higher amplitude of ${b}_{\varphi}({b}_{1})$ . For $z=0.5$ and
${b}_{1}\approx 2$ in particular, the population comprising either all halos,
those with the $33\mathrm{\%}$ lowest or those with the $33\mathrm{\%}$ highest concentrations
have a PNG bias of ${b}_{\varphi}\approx 3$ , ${b}_{\varphi}\approx -1$ and ${b}_{\varphi}\approx 9$ , respectively. Varying the halo concentration can make ${b}_{\varphi}$ very small
and even change its sign. These results have important ramifications for galaxy
clustering constraints of the local PNG parameter ${f}_{\mathrm{N}\mathrm{L}}$ that assume
fixed forms for the ${b}_{\varphi}({b}_{1})$ relation. We illustrate the significant
impact of halo assembly bias in actual data using the BOSS DR12 galaxy power
spectrum: assuming that BOSS galaxies are representative of all halos, the
$33\mathrm{\%}$ lowest or the $33\mathrm{\%}$ highest concentration halos yields ${\sigma}_{{f}_{\mathrm{N}\mathrm{L}}}=44,165,19$ , respectively. Our results suggest taking host halo
concentration into account in galaxy selection strategies to maximize the
signal-to-noise on ${f}_{\mathrm{N}\mathrm{L}}$ . They also motivate more simulation-based
efforts to study the ${b}_{\varphi}({b}_{1})$ relation of halos and galaxies.