In the origin of life, one of the important steps was the first emergence of self-reproducing entities capable of Darwinian evolution. There have been several proposals for the earliest self-reproducing entities, which could be broadly described as consisting of autocatalytic chemical reaction networks confined within compartments that grow and divide to produce new offspring [1]. For such entities to be capable of evolution, they must have certain properties, one of which is the heredity of phenotypic traits [2,3]. The simplest example of a self-reproducing entity that exhibits heredity would be a bistable autocatalytic chemical reaction network in a growing and dividing compartment, which exhibits two growth states with different chemical compositions. Given such a chemical system, how can we test if it will exhibit heredity, i.e., if these states will be stable upon the growth and division of the compartments? Here, we study this question theoretically using deterministic dynamical models of a class of autocatalytic chemical systems that exhibit bistability and investigate how growth and division alter the chemical rate equations. First, we examine a serial dilution protocol for the propagation of the chemical network and show heredity of compositional information only occurs when the time interval between dilutions is below a critical threshold that depends on the efficiency of the catalytic reactions. We then show that these thresholds provide rigorous bounds on the properties of more general growth and division cycles that are necessary for the heredity of the chemical compositional state. Our results suggest that a serial dilution experiment, which is much easier to set up in a laboratory than a general growth and division scenario, can be used to test whether a given autocatalytic chemical system can exhibit heredity of its compositional states. Lastly, we apply our results to a realistic autocatalytic system based on the Azoarcus ribozyme [4,5], and suggest a protocol to test whether the system can exhibit heredity. [1] S. Ameta, Y.J. Matsubara, N. Chakraborty, S. Krishna, and S. Thutupalli, Life 11(4), 308 (2021). [2] R.C. Lewontin, Annu. Rev. Ecol. Evol. Syst. 1(1), 1-18 (1970). [3] P. Godfrey-Smith, J. Philos. 104(10), 489-516 (2007). [4] N. Vaidya, et al. Nature 491(7422), 72 (2012). [5] S. Arsène, S. Ameta, N. Lehman, A. D. Griffiths, and P. Nghe, Nucleic Acids Res. 46(18), 9660-9666 (2018).